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ex.24.10.1.127_255_383.a

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 \(x^{2} + ((70425033346012744527594622521a - 70425033346012744527594622521)b^{4} )x + 3b \)

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

Inertia Polynomial

The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + -196860485231210131548314975448a - 591439414626939433156721435672 x^{47} + (-143926455190527250374176993788a + 174054149062916556661406761880 )x^{46} + 266517720665873991782778474436a - 119344806451329172985022509904 x^{45} + 204349715918512703740299116248a - 83097859178388856783536096636 x^{44} + (552045536642448450206539381808a + 185374621559234273231630332736 )x^{43} + (454323850985532384357119860792a + 133430460953373343603485611116 )x^{42} + -481821013431161050899654601256a - 522906294691122310799349200752 x^{41} + 302968712877725888802760293516a - 597564812215119649560589180388 x^{40} + -357235914098877605506470671352a - 195692110848279355937912921984 x^{39} + 50306634523766525994351556968a - 326738429037209010517420106760 x^{38} + (415934601647077617256262401852a + 12430401900869616133987016904 )x^{37} + (-167094489247782361646673216072a + 354178596753288737077338519324 )x^{36} + -428822453728199297504330638808a - 153742647244942451696570724640 x^{35} + (32393065932282861573068988324a + 324239519524506830976230830784 )x^{34} + 86941488662020250748965635376a - 17564499088371685595655210568 x^{33} + (-55874108092905141830242892880a + 476410549009979073610244261208 )x^{32} + (89368548027315187511046010208a + 188653873180077072459307233280 )x^{31} + (-349978816118338347057307892264a + 43086418953254634043225025840 )x^{30} + 346093973708326847449145192112a - 4490262338725107912768394104 x^{29} + (46464309776692655655193314312a + 366847728213770600769296375440 )x^{28} + (-330488671000820487736996781168a + 312612203887354129571810602944 )x^{27} + (10564825256068009025744465204a + 488128190154380291688563348920 )x^{26} + (-10107700204442836593075274576a + 407394071713531050401265870752 )x^{25} + (-9712941865088278703683765410a + 266332812159096556358860873832 )x^{24} + (-540305163364347148736734074200a + 169474980471030623777708857424 )x^{23} + 404710269356446436922013101752a - 550937891349657936369004411688 x^{22} + (-375353428115592645354514001448a + 455326674083925268342251083320 )x^{21} + (-142535582464467703463446498356a + 96469251918030669845929209280 )x^{20} + (-537054291900254943834432910976a + 502971260245301084246424923888 )x^{19} + -532873914861082448318044839652a - 212026867584103112691644974832 x^{18} + (-417688724234748653440255993920a + 350290846377483819149365056128 )x^{17} + -3779086769919112611609613484a - 609185157674599240250891901576 x^{16} + (164253950890178584091264163280a + 508641915588433966227050804224 )x^{15} + (396746111370044654091411938160a + 441665044911656053371951077504 )x^{14} + -515582862147549073999885313888a - 138411011352595484308845577784 x^{13} + 476910062777367381570150765696a - 19371250775909039676375925304 x^{12} + (-68267706991412287911112212448a + 26867097544478729439017096592 )x^{11} + (22924223574074587757581811360a + 379713180303729972003576959672 )x^{10} + (-495153771302708583244655237768a + 234210607401796112200926242464 )x^{9} + 257528127600132608439858415960a - 355112795602576982622032891456 x^{8} + 515978249915546013182647488912a - 317886447341771014525550090224 x^{7} + 408224440704979106195584248048a - 212640810783594961403299187296 x^{6} + (-438396421591511352789545144024a + 390311618449688179192016262496 )x^{5} + (-583004035989196184849592322688a + 82188106202404445367442576368 )x^{4} + 127544356156735525212300957328a - 86373409019928166498883425216 x^{3} + 422980196439428777942915627072a - 632869592692801551315992798536 x^{2} + -366376593668357501849044399984a - 239966730480057848760131428288 x + 16800387792883497903964970230a - 334106943111334423347062118238 \)
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