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ex.24.10.1.33_67_101.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 \(x^{2} + 253530120045645880299340641075b^{3} x + (63382530011411470074835160269a\cdot b + 253530120045645880299340641075)b \)

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^{ 2 } \\ \chi^A\left(((2a\cdot \mu_3 - 2)b^{2} + 2\mu_3b + 4)c + ((2a + 4)\mu_3 + (2a + 2))b^{2} + ((2a + 3)\mu_3 + (2a + 3))b + 4\mu_3 + 4a - 1 \right) &= i^{ 0 } \\ \chi^A\left(((2a + 4)\mu_3b^{2} + (4a - 2)\mu_3)c + 4\mu_3b^{2} + 4a\cdot \mu_3 + 1 \right) &= i^{ 2 } \\ \chi^A\left((((a + 2)\mu_3 + (2a - 1))b^{2} + (a - 1)\mu_3b + ((-2a - 2)\mu_3 + (4a + 2)))c + (3\mu_3 + (3a - 2))b^{2} + (\mu_3 + (a + 1))b + (2a + 2)\mu_3 + 3a + 1 \right) &= i^{ 0 } \\ \chi^A\left(((2a + 1)b^{2} + (2a + 2)b)\cdot c + (2a + 4)b^{2} + 2a\cdot b + 1 \right) &= i^{ 0 } \\ \chi^A\left((((3a - 1)\mu_3 + (3a - 2))b^{2} + (3a + 3)\mu_3b + ((2a - 3)\mu_3 - 3a + 4))c + ((2a + 3)\mu_3 + (3a + 4))b^{2} + ((2a + 3)\mu_3 + 3a)\cdot b + 3\mu_3 + 4 \right) &= i^{ 0 } \\ \chi^A\left(((-2\mu_3 + (2a - 2))b^{2} + ((a + 4)\mu_3 + 3a)\cdot b + ((-2a + 4)\mu_3 - a + 4))c + ((a + 4)\mu_3 + (3a + 2))b^{2} + (\mu_3 + 1)b + (2a + 2)\mu_3 + 2a - 3 \right) &= i^{ 2 } \\ \chi^A\left(((3a\cdot \mu_3 + 4)b^{2} + (-2\mu_3 + (3a - 2))b + ((-a + 4)\mu_3 + (2a - 2)))c + ((3a + 2)\mu_3 + (2a + 4))b^{2} + ((3a + 1)\mu_3 + 4)b + (3a + 3)\mu_3 + 4a \right) &= i^{ 0 } \\ \chi^A\left((((a + 4)\mu_3 + (2a + 2))b^{2} + (2a\cdot \mu_3 + 4)b + (2a\cdot \mu_3 + 4a))\cdot c + (2\mu_3 + 2a)\cdot b^{2} + (-2\mu_3 + 2)b + (3a - 2)\mu_3 + a - 3 \right) &= i^{ 2 } \\ \chi^A\left((2\mu_3b^{2} + ((3a - 2)\mu_3 + 4)b + ((a - 2)\mu_3 + (4a + 4)))c + (3a\cdot \mu_3 + 4)b^{2} + ((3a + 1)\mu_3 + 4)b + (-3a + 3)\mu_3 \right) &= i^{ 0 } \\ \chi^A\left(((2a\cdot \mu_3 + (2a + 4))b^{2} + ((a + 4)\mu_3 + (2a + 4))b + 4)c + (a - 3)\mu_3b^{2} + ((a - 1)\mu_3 + (3a + 3))b + (4a + 4)\mu_3 - a + 1 \right) &= i^{ 2 } \\ \chi^A\left((((2a + 2)\mu_3 + (3a + 2))b^{2} + ((a + 4)\mu_3 + (2a - 3))b + (2a\cdot \mu_3 - a + 4))c + (2a - 1)\mu_3b^{2} + (a\cdot \mu_3 + 2)b + (4a - 2)\mu_3 + 4a - 3 \right) &= i^{ 0 } \\ \chi^A\left((((3a + 4)\mu_3 + (3a + 4))b + ((a - 2)\mu_3 + (a - 2)))c + ((a - 1)\mu_3 + (a - 1))b^{2} + (-\mu_3 - 1)b + (4a - 1)\mu_3 + 4a - 1 \right) &= i^{ 0 } \\ \chi^A\left(((2a + 1)\mu_3b^{2} + (2a + 2)\mu_3b)\cdot c + (2a + 4)\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} + 382452038323262919676283539244a - 598674030755459596329358279672 x^{47} + (-174745700220257412928811946276a + 455477434563270207205613598836 )x^{46} + -571957320439512414654136973312a - 424275230646977154913260001968 x^{45} + 283952444410258638422530433856a - 489459669298066524151971328604 x^{44} + (-419659084447686313035224534340a + 23014244448372994891806638188 )x^{43} + (-472329333135235188991356299536a + 102782400573873548099397747544 )x^{42} + 184958255604066658300027663948a - 378282654579295166611819508648 x^{41} + (-317344807697225865241311194564a + 426841982345397878603288888712 )x^{40} + -590227875961278132579870746288a - 211777980185495422925677492456 x^{39} + (-436716934875140888013793365828a + 14103677748302540791547987356 )x^{38} + 460952829691644484672737133260a - 111771208263309531667325804708 x^{37} + (-328393073094117894225720918592a + 385569012257478938814663491560 )x^{36} + (-193382534519518314874785469344a + 535897251707210817982530291592 )x^{35} + -276437438714210016288817052224a - 219197372167471052150217028208 x^{34} + (176387575206756988002780078996a + 143746700608003884679491098856 )x^{33} + 597336546502592346506532434404a - 481141822696383475534498476832 x^{32} + (396899473737069247001532904264a + 373943913311157453417906896288 )x^{31} + (307438384829128190040175360452a + 490152947374835111585160000164 )x^{30} + (311693399347463800309629643768a + 421758764438783958847019432088 )x^{29} + -361604465791723246560514844304a - 185344161141552188654121070680 x^{28} + (-4661915483700537420762308184a + 402987086310219834943360952504 )x^{27} + (-187543924857652183037978857320a + 633709727859658441428229994508 )x^{26} + (-620532004488406264329316795696a + 82607900795939897834059861392 )x^{25} + 131129029263898090140993698818a - 545140063204910408186226618668 x^{24} + 605443620832155678901364561768a - 487237943424414283704860535176 x^{23} + (-269061985116992559195267509996a + 326122313874476286624935622320 )x^{22} + (-48813212599190727879494772864a + 613531438059753161075632057520 )x^{21} + (306874936574570493416528051444a + 268159517853693257047616836384 )x^{20} + 109178849321328907025283917236a - 610928010950995345235751236216 x^{19} + -287846976220477777182258526584a - 286605962394125966913421349136 x^{18} + -41571522704328992745003619560a - 305585646019791920251067879848 x^{17} + (-347318447663222798660137886144a + 345009599907319006880674781832 )x^{16} + (-505033273808938427975277395872a + 379764190472390492276311813696 )x^{15} + 322735909441073720518300961948a - 44540591545127007602398106336 x^{14} + -31959977897555213865853410620a - 342149208780002210863931835616 x^{13} + (-374687905602846553184828253976a + 621532540572449975950975553324 )x^{12} + 629894932409995978981501390584a - 438950598723838467594369703808 x^{11} + 415706778246431317032724326800a - 171868623010334433554523924512 x^{10} + 413763286373566895851147044448a - 338642001264861775894823185736 x^{9} + 575874541769058049031534833072a - 326970166312778645341687533120 x^{8} + 575130532177156273722359610352a - 38638169419669364501487450544 x^{7} + (-27139572954463897811779588108a + 52243799498921080159098164928 )x^{6} + (-569340782516544253569405800600a + 473155211462114928515262151792 )x^{5} + (-356049187511221450645188191152a + 399598285770399626215256060832 )x^{4} + 83753602990732494535186169144a - 519325756986080658671497698016 x^{3} + (358480554934265155442487354116a + 406562875241992572259062050680 )x^{2} + -495121469566689334464137590184a - 100495385675541547661133107344 x - 563904203811746504605473003212a + 83476010314576974224171488090 \)
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