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ex.24.10.1.127_255_383.d

Base Field
\(F = \) 2.1.2.3a1.2 \( = \mathbb{Q}_{ 2 }(a) = \mathbb{Q}_{ 2 }[x] / (x^{2} + 5\cdot 2 )\) 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 \(x^{2} + 253530120045645880299340641075b^{4} x + b \)

Underlying Character

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

Order
4
Conductor exponent
18
Values on generators of \((\mathcal{O}_K/\mathfrak p^{ 18 })^\times/U_{\mathfrak{p}^{ 18 } }\) :
\(\begin{array}{l} \chi^A\left((-2\mu_3b^{2} - \mu_3b)\cdot c + a\cdot \mu_3b^{2} + 1 \right) &= i^{ 2 } \\ \chi^A\left(a\cdot \mu_3c - 2\mu_3b + 1 \right) &= i^{ 3 } \\ \chi^A\left(-2\mu_3b^{2}c + 1 \right) &= i^{ 2 } \\ \chi^A\left((-\mu_3 + 3)c + -\mu_3 + 3 \right) &= i^{ 0 } \\ \chi^A\left((\mu_3b^{2} + 2a\cdot \mu_3)c + 4\mu_3b + 4\mu_3 + 1 \right) &= i^{ 2 } \\ \chi^A\left(((2a\cdot \mu_3 + (2a + 4))b^{2} + ((2a - 1)\mu_3 + (a + 2))b + ((2a - 3)\mu_3 + (a + 4)))c + ((2a - 3)\mu_3 + 3)b^{2} + (2a\cdot \mu_3 + (2a - 2))b + (a + 2)\mu_3 + 3a - 3 \right) &= i^{ 1 } \\ \chi^A\left((b^{2} + 2a)\cdot c + 4b - 3 \right) &= i^{ 0 } \\ \chi^A\left((((3a + 2)\mu_3 + (3a + 2))b^{2} + ((3a - 2)\mu_3 + (3a - 2))b + ((2a - 2)\mu_3 + (2a - 2)))c + ((a + 1)\mu_3 + (a + 1))b^{2} + ((a + 1)\mu_3 + (a + 1))b + (3a + 1)\mu_3 + 3a + 1 \right) &= i^{ 0 } \\ \chi^A\left((4b^{2} + (4\mu_3 + 2)b - 2\mu_3)c + (2a\cdot \mu_3 + (2a + 4))b^{2} + (\mu_3 + (2a + 1))b + 1 \right) &= i^{ 0 } \\ \chi^A\left((2a\cdot b + 2a\cdot \mu_3)c + (2a\cdot \mu_3 + 2a)\cdot b^{2} + (-3\mu_3 + (2a + 1))b + 1 \right) &= i^{ 0 } \\ \chi^A\left(((a - 2)b^{2} + ((a + 2)\mu_3 + 4)b + ((2a - 2)\mu_3 + (2a - 2)))c + (3a - 1)\mu_3b^{2} + ((a - 3)\mu_3 + (a + 1))b + 2a\cdot \mu_3 + 3a - 1 \right) &= i^{ 0 } \\ \chi^A\left((-2b^{2} - b)c + a\cdot b^{2} + 1 \right) &= i^{ 0 } \\ \chi^A\left((((2a - 2)\mu_3 + 4)b^{2} + ((2a - 2)\mu_3 + (2a + 2))b + (2a + 2))c + ((3a - 1)\mu_3 + 2a)\cdot b^{2} + ((3a - 3)\mu_3 + (a + 1))b + 4\mu_3 + a + 3 \right) &= i^{ 0 } \\ \chi^A\left(a\cdot c - 2b + 1 \right) &= i^{ 0 } \end{array} \)

Inertia Polynomial

The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + -148448310797480409538846019856a - 250306187844320568896208682088 x^{47} + (254447597113013539572040298828a + 594576935500353636191932094336 )x^{46} + 309786780377151911565322219948a - 625589276032881845939439321312 x^{45} + 612977623978067396181217830556a - 383411982201342207827595042996 x^{44} + (-53188894620931228701419221936a + 223555906183880693488287791248 )x^{43} + -316209983598474093377464806636a - 292243528661541263797703226340 x^{42} + 331930993051657243286721439576a - 95521430392510857290660390776 x^{41} + (59917510495544423910014657208a + 173354771086337899200568069100 )x^{40} + (-459897714397970277984736191208a + 156612267573831173368068490832 )x^{39} + (-578957182776116293068908159492a + 96164097829870345112817029136 )x^{38} + (-286722030745981465043256078412a + 366521095349384925601477454912 )x^{37} + (139908716362353729787339275512a + 518585900195543703721117070444 )x^{36} + 151337016207100780058818665928a - 567456369086077001893758221840 x^{35} + -467909716464021756192281799348a - 410123628218689626676993086416 x^{34} + 3471319385265353734400446648a - 201166455828564725736815284168 x^{33} + (132620579519027349687345333840a + 387507581349987396146566992992 )x^{32} + (272006352712915710306941313376a + 527280420722176896457501385088 )x^{31} + (408111295468184233315995662344a + 312402367121867597775276112160 )x^{30} + (235770253839509010577492931032a + 294906983223417151599615497752 )x^{29} + 503114926862920769134070977816a - 575244945667948870550353050960 x^{28} + -304300939794311381050545730880a - 491202228674305903711546595072 x^{27} + (-205906661064972208590614050764a + 354508673002444520147274723000 )x^{26} + (-563603818943293627544186005584a + 601064507035148130919840357792 )x^{25} + (558321019633589761069177457246a + 485462758888673751730687575016 )x^{24} + 369845549422769646183950620360a - 327139874230470130453009687968 x^{23} + 207816073468078163674151529800a - 354331057021973514746093459496 x^{22} + (-65643935892713648212059981696a + 229496396909686750049677593976 )x^{21} + (-132942253675366684714203334716a + 633599714645306574331600965464 )x^{20} + (588670293331853163224664648864a + 253292669411966701193190144848 )x^{19} + -463756121048817277646841828324a - 382556488047569888100288607400 x^{18} + (269776627635639287358966401512a + 421655770873608232172069642656 )x^{17} + (624850092813020301540509960780a + 514435042291541485884865810096 )x^{16} + (-392505640621799348579224940016a + 99494673671705803558324654208 )x^{15} + 372895916436050165474344321272a - 607447327183705033436794710504 x^{14} + (71949847572279726961320694896a + 290322173057335368701179685928 )x^{13} + (444647703188985772817851575616a + 401913813787073273990737824248 )x^{12} + -42178803330207978496288677088a - 436499181219029538510464179696 x^{11} + 562545055017997525073218241560a - 480022139031806393206928661928 x^{10} + (388616683391816956934266250200a + 618192224128185436446115472240 )x^{9} + (68834725356924140624496921232a + 543264705758061023381941829584 )x^{8} + -268159132006782159999204174528a - 308116064058905444057868341360 x^{7} + 523373603846177599708957595904a - 316085126436822834468122155904 x^{6} + (101100584999833879589218520920a + 538886756329291679555809755984 )x^{5} + (422083067480164531917067864640a + 347128650916776602623610418192 )x^{4} + (-396385795683756599591980374768a + 263691562553274796543013614816 )x^{3} + (432178280209038594150765753784a + 396312922102768503078211360232 )x^{2} + (-630274061343810448539693798512a + 64701354318812467450396712288 )x + 532362118804953164518448550432a - 385296978002671271676701293602 \)
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