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ex.24.10.1.33_67_101.a

Base Field
\(F = \) 2.1.2.2a1.2 \( = \mathbb{Q}_{ 2 }(a) = \mathbb{Q}_{ 2 }[x] / (x^{2} + 2 x + 3\cdot 2 )\) View on LMFDB ↗
Description
exceptional, SL(2,3)
Construction
Extension to \(I_F\) of \( \tau = \operatorname{Ind}^{I_K}_{I_L} \chi \), with \(L, K\) and \(\chi\) as below
Semistability defect
\( e = 24\)
Conductor exponent
\( v(N) = 10\)
Character Order
4

Triply Field

The inertial type \(\tau\) becomes an induction (triply imprimitive) over: \( L = F(\mu_3, b) \), with \(\mu_3\) a root of \(x^2+x+1\) and \(b\) a root of \(x^{3} + a \)

Inducing Field

The inertial type \(\tau\) becomes reducible over \(K = L(c)\), \(c\) a root of \(x^{2} + ((70425033346012744527594622521a - 70425033346012744527594622521)b^{3} )x + ((-105637550019019116791391933781a - 211275100038038233582783867562)b + (-140850066692025489055189245042a + 140850066692025489055189245041))b \)

Underlying Character

Character \(\chi^A:\mathcal O_K^\times \to \mathbb C^\times\) with the following properties:

Order
4
Conductor exponent
20
Values on generators of \((\mathcal{O}_K/\mathfrak p^{ 20 })^\times/U_{\mathfrak{p}^{ 20 } }\) :
\(\begin{array}{l} \chi^A\left(\mu_3c + 1 \right) &= i^{ 0 } \\ \chi^A\left((((2a + 2)\mu_3 - 2)b^{2} + (4\mu_3 - 2)b - 2\mu_3 + 4a + 4)c + ((3a + 3)\mu_3 + 2a)\cdot b^{2} + ((a - 3)\mu_3 + (a - 3))b + (2a + 4)\mu_3 - a + 1 \right) &= i^{ 0 } \\ \chi^A\left((((a + 4)\mu_3 + (2a + 2))b^{2} + ((2a + 4)\mu_3 + 4)b - 2a\cdot \mu_3)c + (2\mu_3 + 2a)\cdot b^{2} + (2\mu_3 + 2)b + 3a\cdot \mu_3 - a - 3 \right) &= i^{ 2 } \\ \chi^A\left(((3a\cdot \mu_3 + (a + 3))b^{2} + ((2a - 2)\mu_3 + (3a - 1))b + (a\cdot \mu_3 + 2a))\cdot c + (\mu_3 + 3)b^{2} + ((3a + 1)\mu_3 + (2a - 1))b + (-a + 1)\mu_3 - a - 1 \right) &= i^{ 2 } \\ \chi^A\left(((2a + 1)b^{2} + 2b)\cdot c + 2a\cdot b^{2} + 2a\cdot b + 1 \right) &= i^{ 0 } \\ \chi^A\left((((2a + 4)\mu_3 - 3)b^{2} + ((2a + 4)\mu_3 + (a + 4))b + ((4a + 4)\mu_3 - 1))c + ((2a + 4)\mu_3 + (2a - 3))b^{2} + (4\mu_3 + 3a)\cdot b + -2a - 1 \right) &= i^{ 0 } \\ \chi^A\left(((2a + 2)\mu_3b^{2} + ((3a + 4)\mu_3 + (2a + 4))b + ((3a + 4)\mu_3 + 4a))\cdot c + ((a + 2)\mu_3 + 4)b^{2} + ((2a + 3)\mu_3 + 4)b + (-2a - 3)\mu_3 \right) &= i^{ 0 } \\ \chi^A\left((3a\cdot \mu_3b^{2} + (a + 4)\mu_3b + 2a\cdot \mu_3)c + (a - 2)\mu_3b^{2} + (2a + 1)\mu_3b + (-2a - 1)\mu_3 \right) &= i^{ 0 } \\ \chi^A\left(((-2\mu_3 + a)b^{2} + (4\mu_3 + (3a + 4))b + ((-a + 2)\mu_3 - 2a + 4))c + ((a + 2)\mu_3 - 2)b^{2} + ((2a - 1)\mu_3 + 4)b + \mu_3 + 4 \right) &= i^{ 2 } \\ \chi^A\left(((4\mu_3 + 4)b^{2} + (2a + 4)b + (4\mu_3 + (4a + 4)))c + ((a + 2)\mu_3 + (a + 4))b^{2} + (\mu_3 + (2a - 3))b + (-2a - 2)\mu_3 - 2a - 1 \right) &= i^{ 2 } \\ \chi^A\left((((2a - 2)\mu_3 + (2a + 2))b^{2} + (3a\cdot \mu_3 + (a + 4))b + (3a\cdot \mu_3 - a + 4))c + (-\mu_3 + 3)b^{2} + ((a - 3)\mu_3 + (a + 1))b + (-a - 3)\mu_3 + 3a - 3 \right) &= i^{ 0 } \\ \chi^A\left(((a + 2)b^{2} + (2a - 1)b + (2a - 2))c + (2a - 2)b^{2} + (3a + 2)b + 1 \right) &= i^{ 0 } \\ \chi^A\left((((a + 2)\mu_3 - 2)b^{2} + ((2a + 4)\mu_3 + (3a + 4))b + (2\mu_3 + (3a - 2)))c + ((2a - 1)\mu_3 + (a - 2))b^{2} + ((3a - 1)\mu_3 + (a + 3))b + (-2a + 2)\mu_3 - a + 3 \right) &= i^{ 0 } \\ \chi^A\left(((2a + 1)\mu_3b^{2} + 2\mu_3b)\cdot c + 2a\cdot \mu_3b^{2} + 2a\cdot \mu_3b + 1 \right) &= i^{ 1 } \end{array} \)

Inertia Polynomial

The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + (392689259821872908561012493804a + 543726715829877650579450068896 )x^{47} + -480501888438904084859720107152a - 588221752381119549309786628332 x^{46} + (-481114760433774140776063499696a + 383508276234378964846115451384 )x^{45} + 599394968325657710133385682084a - 146068926365925118237731661764 x^{44} + (386413160062965692385765944356a + 515690958860092804248708263452 )x^{43} + -221045952913182544452832239008a - 124232887939304521133008099184 x^{42} + -521836274203418410851514333452a - 568913077965505088499828183184 x^{41} + 287845367723579331671610113304a - 11019547604237049983180028920 x^{40} + 32575317161505268666435926448a - 205369916859282677688401450184 x^{39} + -560984225195737768183191760404a - 308619659397416541509171236116 x^{38} + 344705998990540961085883381204a - 576422254061553798188980390284 x^{37} + -228985922359929827643171120600a - 493741305893600390561893846040 x^{36} + (275279245435541045567021045624a + 480686453919608060077197769280 )x^{35} + (112743400620524399629987726100a + 400018717910837875686949491392 )x^{34} + -590536356498941496491019956116a - 531897349742613320452940146104 x^{33} + -75881840566438681341403670800a - 437219885774748412413660389688 x^{32} + (12689509991265071661836354008a + 251609238429846323762136609264 )x^{31} + (-570101702799810402141657038304a + 519849629494532587466717714196 )x^{30} + (602604271135498047303781822620a + 228227048875848446295350049752 )x^{29} + (262391630944844753393789266008a + 131249220251240211993524411016 )x^{28} + (-356691989401554165235967493896a + 16939611077311384455205940504 )x^{27} + 265593755879245822074324880664a - 555326306840914435923950022148 x^{26} + (-332918524281189326091927564904a + 623916962829798739591147276184 )x^{25} + (-257664585468911363446025178042a + 266551494898497691024747991508 )x^{24} + (440995557930173221842959827688a + 105601650282094028683786223784 )x^{23} + (322850258261648275668088017468a + 155752503214529995502470544984 )x^{22} + (103329060086176779741695111032a + 425414676730471966761019130272 )x^{21} + -619622573204131638582636283460a - 577179509754351175796727616312 x^{20} + (-403419198007458185477204345588a + 20905266347493448364304450776 )x^{19} + (-532738513020383128299793356496a + 113622211909436265525803028800 )x^{18} + (405467182593644143332105821352a + 622885278632695858385633809384 )x^{17} + -548494318683324481709593382368a - 461377883702985421527316513648 x^{16} + -428639449093581727947815735216a - 186537709818225801386422803760 x^{15} + (-1355646197050917463472319772a + 165072612458381780482640894624 )x^{14} + (282243096459982981408822544572a + 211785064157998610056547189280 )x^{13} + -588483774759739672260688964592a - 26006897198225562699994626508 x^{12} + (-462171215875189180011331274504a + 472481705002422720410877178624 )x^{11} + (-238176646428493731433640193488a + 27245151952317282535614789080 )x^{10} + -147677704255040354376705075448a - 27019715569420839198236779160 x^{9} + (135831290121833954053810273032a + 129178774191957156661655625112 )x^{8} + (-506162413718176814770002023216a + 88989555870921688971395822816 )x^{7} + (-563001781784889708812911533516a + 91638336620905498278086158168 )x^{6} + 13119601652984809384064853800a - 277746204862738019963865390456 x^{5} + 547368848972343468532920247680a - 353022054652973897932562669264 x^{4} + -407845793697337267135385127304a - 424532263747157292082913807200 x^{3} + -126001439408652128200895171092a - 532247872259575843569984690344 x^{2} + 244573355341638423344415377632a - 176662061292792933022254079584 x + 505571265107742619189922786906a - 544345061875870561657857793266 \)
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