← Back to 2.1.2.2a1.2

ex.24.10.1.31_63_95.c

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 \(x^{2} + ((70425033346012744527594622521a - 70425033346012744527594622521)b^{3} )x + ((-140850066692025489055189245042a + 140850066692025489055189245041)\mu_3 - 140850066692025489055189245042a + 140850066692025489055189245041)b \)

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^{ 3 } \\ \chi^A\left(((2a + 2)b^{2} + (-2\mu_3 - 2)b + ((4a - 2)\mu_3 + 4))c + ((3a + 1)\mu_3 + (2a - 1))b^{2} + ((3a + 3)\mu_3 + (2a + 2))b + 2a\cdot \mu_3 + 3a - 3 \right) &= i^{ 0 } \\ \chi^A\left((((3a - 2)\mu_3 + (a + 4))b^{2} + (4\mu_3 + 2a)\cdot b + 4a\cdot \mu_3)c + ((2a + 2)\mu_3 + (2a - 2))b^{2} + 4b + (a + 4)\mu_3 + a - 3 \right) &= i^{ 0 } \\ \chi^A\left(((2a + 1)b + (2a - 2)\mu_3)c + (a - 2)b + 1 \right) &= i^{ 0 } \\ \chi^A\left((((2a - 1)\mu_3 + (2a - 1))b + (2a - 2))c + ((3a + 2)\mu_3 + (3a + 2))b + 1 \right) &= i^{ 0 } \\ \chi^A\left((((2a - 3)\mu_3 + (a + 1))b^{2} - 3b + (-a + 3)\mu_3 - a)c + ((a + 3)\mu_3 + 3)b^{2} + ((a + 4)\mu_3 + (2a + 1))b + (3a - 3)\mu_3 + 4a \right) &= i^{ 2 } \\ \chi^A\left((4\mu_3b^{2} + ((2a + 4)\mu_3 + (3a + 4))b - a\cdot \mu_3 + 4a)\cdot c + 4\mu_3b^{2} + (4\mu_3 + (2a - 3))b + (-2a - 3)\mu_3 \right) &= i^{ 0 } \\ \chi^A\left((((3a - 2)\mu_3 + 4)b^{2} + (a - 2)b + ((4a + 2)\mu_3 + (2a - 2)))c + ((a + 1)\mu_3 + (2a - 1))b^{2} + ((3a + 3)\mu_3 + 2)b + 4a\cdot \mu_3 + a - 3 \right) &= i^{ 2 } \\ \chi^A\left((((a + 4)\mu_3 + (a + 4))b^{2} + (2a + 4)b + (-2a - 2)\mu_3)c + (-2\mu_3 - 2)b^{2} + (2a - 1)b + (4a + 3)\mu_3 \right) &= i^{ 0 } \\ \chi^A\left(((2a\cdot \mu_3 + 2a)\cdot b^{2} + (2a + 4)b + (4\mu_3 + (4a + 4)))c + ((2a - 2)\mu_3 + 4)b^{2} + ((2a + 1)\mu_3 + 2a)\cdot b + (2a - 2)\mu_3 - 2a + 3 \right) &= i^{ 2 } \\ \chi^A\left(((4\mu_3 + (2a + 4))b^{2} + ((a + 2)\mu_3 + (3a + 4))b + ((-3a + 4)\mu_3 + 2a))\cdot c + (2a\cdot \mu_3 + 4)b^{2} + (2a + 1)b + \mu_3 + 4 \right) &= i^{ 0 } \\ \chi^A\left(((-\mu_3 - 1)b^{2} + (2a + 2)b + 4\mu_3)c + ((2a + 4)\mu_3 + (2a + 4))b^{2} + 2a\cdot b + 1 \right) &= i^{ 1 } \\ \chi^A\left((((3a + 4)\mu_3 + 3a)\cdot b^{2} + ((2a + 4)\mu_3 + (3a + 4))b + (2a\cdot \mu_3 + 4a))\cdot c + (4\mu_3 + 4)b^{2} + (4\mu_3 + (2a + 3))b + (-2a - 1)\mu_3 + 4a \right) &= i^{ 0 } \\ \chi^A\left((\mu_3b^{2} + ((2a - 2)\mu_3 + (2a - 2))b + 4)c + (2a + 4)\mu_3b^{2} + (2a\cdot \mu_3 + 2a)\cdot b + 1 \right) &= i^{ 0 } \end{array} \)

Inertia Polynomial

The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + (593292185660045016857010310740a + 578955631418283290596365249720 )x^{47} + -459781666193787367905170171476a - 514785152573119932541896966432 x^{46} + (-446282352022849051084164005040a + 377028211367459529975539943960 )x^{45} + 285997836332879504037823928460a - 300785336137331765979454579184 x^{44} + (-447957048235840040534091232712a + 425206801574898621102149947228 )x^{43} + (-348770079014156958513314250352a + 560827011102693274568325800128 )x^{42} + 379262869695722087945332740232a - 126279011444490273892263693152 x^{41} + -111680692173089293262601115200a - 309182548993419256758840881776 x^{40} + -380260787960804040381274841200a - 109032277402089760202165459488 x^{39} + (461314879187239202943685016016a + 537884214986163992627301419332 )x^{38} + (-196711453806038162135624567340a + 386540904214131172572662953696 )x^{37} + (186853963004667539963374007788a + 631703350592269902629775214624 )x^{36} + (189614572456772683183764914456a + 519628744838945641068195061720 )x^{35} + -367903722020747155992826679516a - 554512720571846709932309230336 x^{34} + (-419983573053749444288204492472a + 144891104748854490709772906024 )x^{33} + (-574157068104062803938106675196a + 414073389943386622641790696808 )x^{32} + -314300540377878187564466034140a - 81604361379114627269755358928 x^{31} + -77855263323219460286088329740a - 102637564117429974335276212780 x^{30} + (519398192884049684101191476752a + 555296558957096558728646223256 )x^{29} + (346810630379055521392707197344a + 103738395326408750058326843920 )x^{28} + -91075962874493628025766365584a - 318973891502837787417231989568 x^{27} + -419658106503377582070143337880a - 295450306084273355130692525936 x^{26} + -357483010210646354710108559888a - 339748434218780856745160799304 x^{25} + -204611675588688055789651767866a - 10642696602276878382676956560 x^{24} + 543805664351982473635397137792a - 86117074024637450601134791928 x^{23} + (-503341148949458415984093954536a + 595330647183550335511036813336 )x^{22} + (485571634706775727851956593384a + 174849752772708305271250521888 )x^{21} + -353417376652111700682941358552a - 322847206525763159996501572912 x^{20} + 287092694219534159714319355948a - 237926357159871725431265638368 x^{19} + 390204540262040608842828184856a - 490712273013135156428620088640 x^{18} + -426233570357001819257845744192a - 331788655640218939098655498912 x^{17} + (-338542002611192391029833733336a + 575987887812558016272552331936 )x^{16} + -420989848264402300857576683344a - 237820828511760435904226421648 x^{15} + (578111443778614403958940360036a + 341610295958331227183324135792 )x^{14} + (75034144178151739164698874736a + 536945972078995959145014433592 )x^{13} + (358894696828147262947237019248a + 599284813997205784469415858948 )x^{12} + (-324170736985637323145352649192a + 163656817087025102624849231776 )x^{11} + (-70787856023950786287567445768a + 155254266746157549569438761480 )x^{10} + (-295159453691830298694945572600a + 274458173247078416350438719088 )x^{9} + (-186122448326079091740973135648a + 517835237416630178056248452168 )x^{8} + -123981498226902151816649359768a - 37170516835790760136737047736 x^{7} + (469762126501131642606012359996a + 577060539537336011144094914344 )x^{6} + (192600253925675056397916752888a + 64331259936818653225647268000 )x^{5} + (339845113433094110850885690688a + 574122620657415011234918929552 )x^{4} + (-340714569079234908570328950816a + 362751668516363922303175587136 )x^{3} + (-579043367882758232548544488128a + 18421327989999356964575253136 )x^{2} + 175184429982347426007818779632a - 88483256390445437760899349552 x + 542510237983098491523317824678a - 363104443973657515913635321366 \)
← Back to 2.1.2.2a1.2 Summary