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ex.24.10.1.131_259_387.b

Base Field
\(F = \) 2.1.2.3a1.1 \( = \mathbb{Q}_{ 2 }(a) = \mathbb{Q}_{ 2 }[x] / (x^{2} + 2 )\) 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 \(x^{2} - b^{4} x + 3b \)

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} + -625634798739121866452036941616a - 608134777612697930571714073608 x^{47} + 593320728028940152187065395252a - 556351774229874307357896223728 x^{46} + (369078787001323360382973792036a + 508040014249484352157029923760 )x^{45} + -64094947770808867396783192092a - 515657092400529557045991062636 x^{44} + (-570107164351378002862002997848a + 260768447217692636688581778176 )x^{43} + 23032098134264480422691342108a - 444657537723861671054925052228 x^{42} + 243028362122580315018050737024a - 283687876577484186953703754104 x^{41} + (316264035370401830574174843840a + 124462308432597902729679744924 )x^{40} + -598225602601437078827878420568a - 582412738060102244760268339232 x^{39} + (-488484194574157881331692264628a + 276915284802162287840883729784 )x^{38} + (42709099536916927117780115068a + 114018418925546214785548077000 )x^{37} + -427588493778068769298425000880a - 338139358082904992368571063252 x^{36} + 118374960304113320118555356016a - 373569018318197091543067087952 x^{35} + (562310412547371821550579105476a + 356803885820424128514815542648 )x^{34} + 380781257038934550311259349328a - 351495905117392107921714394808 x^{33} + -138062790326957729515643735548a - 164517407105828559552435616920 x^{32} + (-258910823966866540339635579072a + 41531520664540220479254569248 )x^{31} + 240688219790979971093865876152a - 276960852153906681528060958160 x^{30} + (-185982611004923994450283285344a + 11595356356992368859215129000 )x^{29} + (-395351082733076190342695018172a + 322191244650904822527485287896 )x^{28} + (568904831272273740941627309856a + 56559404105649813655823182080 )x^{27} + -261898055928754114302658985236a - 626652803800824790117263818120 x^{26} + (248410571233449445420322712968a + 548254669361788420833820732184 )x^{25} + (-434934695866623356523376307574a + 597042388719485013213961958228 )x^{24} + (617891511601372554558545240712a + 134841337811478880142496510864 )x^{23} + (239537141888067210044555826648a + 147992780317844141483739340008 )x^{22} + (633378544731744966449304813656a + 5665635715923549471568599480 )x^{21} + (505465519762769025483185492380a + 408319580907759180236682627472 )x^{20} + (-292818511618460202821171378256a + 150765429431964820143874292336 )x^{19} + -154732864168236194165711064324a - 14270468864105112991617830496 x^{18} + -426637628301813321809981364600a - 598744375643340094925783056256 x^{17} + (396012930840850856634947168452a + 426412005589325053432284762648 )x^{16} + (-514368164703382296132449129312a + 256729467735619898981211364992 )x^{15} + (-504071814192189302912196959416a + 405698232968367457419149133400 )x^{14} + 94785971906775831510887283456a - 633728876570515504540936335432 x^{13} + (-300142395418189035241607994888a + 245993875766085395896175631416 )x^{12} + 21131002130463145341458869856a - 460453655791911745505991626336 x^{11} + 147202206258886035037617276728a - 174991364776793604910146664328 x^{10} + 540690189133063921638288754616a - 568144791529811322655018347280 x^{9} + -368221551387297552964201519696a - 273690549982717980093063131208 x^{8} + (489164487636111548906075046144a + 614979294805842009544520651472 )x^{7} + (203935524524380848827372633856a + 503463115709865824018253164928 )x^{6} + (570948127837049329110128457896a + 507025588001127616996110874736 )x^{5} + (-193467206183394810544288630752a + 216729748325514119275857412488 )x^{4} + (595061891803682702987912150640a + 605553867851459546279244929248 )x^{3} + 66993128115410737768589695792a - 344330232194671546644258334232 x^{2} + -259651655292064300732705137464a - 246830318894270566361877711232 x + 373429711117860822427404093604a - 557217570620757685053313431590 \)
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