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ex.24.10.1.131_259_387.b

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 \(x^{2} + 253530120045645880299340641075b^{4} x + 3b \)

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

Inertia Polynomial

The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + 190187233347240206072886582352a - 592791236836087080759662173608 x^{47} + (-289576887540526953330523238940a + 502008769811934493514458169024 )x^{46} + (-417794117293609646106924595612a + 38692761472691144795314131632 )x^{45} + -271760973271591464696437343068a - 223873642283609608741705346380 x^{44} + (-323397735038608182216366334008a + 206428949029205647042917996480 )x^{43} + -160334342295891658405144058196a - 476219278888108517346336184788 x^{42} + 290793577901622146779230924944a - 576206731558699983274704092824 x^{41} + (464803713708243146013195186496a + 388621142450050518090385373068 )x^{40} + (-361933345942325844746502043128a + 305438588265954708983071752672 )x^{39} + -400589549316871639479652141444a - 529854791964594975386174914024 x^{38} + -360232980422735075399034697748a - 468549321951017240326378030648 x^{37} + (379763808344652155371419585040a + 354030602341651088322298357196 )x^{36} + 170562256039563501574101871440a - 345235101492369775455346021648 x^{35} + (-588272518979790945424839976892a + 617854282221511493781154565624 )x^{34} + 318957488331237512816409871152a - 2804610158244679998489187544 x^{33} + -299658159078122863114122542092a - 122186058660452972433822778584 x^{32} + (310425593505105730715668698048a + 529561345426384026786862407200 )x^{31} + (-196415977400885032090550154664a + 185582489866216382510576007856 )x^{30} + (490632158327966439765366681040a + 279702271340389108866141627048 )x^{29} + (554590977905003149516978811252a + 345593163739776573595919388408 )x^{28} + (-328159668280620551637827963680a + 233648050478160307064923236864 )x^{27} + -72411766247025822160035921444a - 5989472516830113948606114136 x^{26} + 65054555378413034444320932744a - 409126146673582692771157734440 x^{25} + 588147866538470074646930925850a - 417257994111182305156964721884 x^{24} + (329314084716092347590230608616a + 497463102730129619026503835536 )x^{23} + (-135062436555442459826792941496a + 260049270507828706176818372488 )x^{22} + -555015653556415566609318019784a - 101741492699453120213524063656 x^{21} + (-373241372396663806928410069604a + 146482194202526044353157442160 )x^{20} + (-582685039973221157503289866128a + 558748000364160329507206695408 )x^{19} + -464170901181288007356718390676a - 570031428915854036133507058208 x^{18} + 38202898174939529540350729864a - 516896332256204734181520318624 x^{17} + -594816777906531641029423420620a - 176859438216982356338600094952 x^{16} + (632176593041152903289739893984a + 253211156910066827680648142592 )x^{15} + (-430454194072673305455956949688a + 558064529896736654402844968984 )x^{14} + -423069882447631687364496389248a - 199273038634895517729739647688 x^{13} + (-153652850792557813973634820616a + 434506715891695751495541510872 )x^{12} + (-506581349615045790848342869344a + 493649658071571877351689882080 )x^{11} + (375561550776585651898883402392a + 230139497544737381443583229400 )x^{10} + (478420331997858734748917592792a + 207625779299960520707650468144 )x^{9} + 429428955787470513343921884304a - 153013517388584642024888664968 x^{8} + (29500443060304408896755637888a + 385326190757554542709554053136 )x^{7} + (17592627845836424659900927616a + 346170887498595207001042898304 )x^{6} + (-217444584998901689946892258744a + 529570925707831899205617865616 )x^{5} + 120836251035613475372645574080a - 188404948693114173151155683384 x^{4} + 52227414957168875282069982704a - 590590170610124747990854525984 x^{3} + 474971891951011896447531684544a - 396811532676946920887736129848 x^{2} + (-348963252414549249040577073784a + 354743110348483241492579486720 )x - 417530054426214571292297954284a + 267479756225391279393339479090 \)
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