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ex.24.10.1.127_255_383.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 \(x^{2} - b^{4} x + b \)

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

Inertia Polynomial

The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + (426620235925232049872189753456a + 245824284014678529528181989112 )x^{47} + (-441838969901912776491755594604a + 543736941915448900614573655376 )x^{46} + (43198346805329681614040371948a + 514418949628232053266620818512 )x^{45} + -115483857758759273829448217736a - 608724123955754050032891659932 x^{44} + (299650957772514093387800631296a + 154237727547775269549163966320 )x^{43} + (-58444425897828877331298524548a + 441950285445818965167579001676 )x^{42} + (-575901974669948518675168565448a + 493012225239454074466955256720 )x^{41} + (-225615275301911311618880662160a + 1570066194282055181074438092 )x^{40} + -493769895742391889647163007208a - 532900943183550517904989605664 x^{39} + (408846459165568043067646278908a + 562111592821207980685637166336 )x^{38} + 380243244587772625825980505324a - 251839831858898020430094924208 x^{37} + (-52181163632453681177941742204a + 172087676544800532957930714580 )x^{36} + (-302760366007054734184295439624a + 7464192529865023428706577216 )x^{35} + -519626146129562756985808638300a - 96855333489126123209514392552 x^{34} + (-141279366145230599338014593288a + 179993523848762781610726976624 )x^{33} + (143076350122991732938478507972a + 335426576658150241203724607960 )x^{32} + 239152646720603337757998424928a - 500660072097835922617221024416 x^{31} + (-591633362325595856336888106304a + 577357412732359353248402223072 )x^{30} + (-480108275561412852552881696728a + 261808765291656790127814814904 )x^{29} + 105951622567257128354400226796a - 100949117504812380477999985024 x^{28} + -571387062400134193490889444112a - 191909858372730730722259552816 x^{27} + 112466168828276738364048216228a - 213682782392636732857608935728 x^{26} + (-323742255010709662264004830776a + 313750604112713833252335593152 )x^{25} + (414470882306876218209679751486a + 19248280969560532101819817616 )x^{24} + 243550121717033435878441485576a - 591036349115715551815660990144 x^{23} + (533036911338864661224103004624a + 525875588279621910893447696376 )x^{22} + (-245960751849529604818303707872a + 551517375485673704203859327608 )x^{21} + -175151850026730243746327126788a - 602924757318843999412220546640 x^{20} + (469579699044863478331712270096a + 112772107734322968678041948240 )x^{19} + (42801956499237704637163940668a + 417361093003380278902152692632 )x^{18} + (424198902694050783476700912592a + 121425513087915950244279543936 )x^{17} + 202305782369295003969065826444a - 617134393906304517963908978848 x^{16} + (37664294627395945328584019488a + 89030014922979500721201248192 )x^{15} + -506591786166704906117176741960a - 337511017397219369114360414776 x^{14} + (-2234738389394467186889272640a + 278886381534090986287812317112 )x^{13} + -179365689840932137137428517928a - 224849562522028655869106616736 x^{12} + (-428689696444692522147214443792a + 325755858237356123867862992336 )x^{11} + (270958010312484961898852442224a + 406024668254094237471151787960 )x^{10} + 535017446538991211594022333936a - 480177896613332395486085642416 x^{9} + (568795038151124242342702473032a + 548409166698576838496125893784 )x^{8} + (-126147430347258551656608184672a + 267807520167843901380718624912 )x^{7} + (130080026134268553059521900400a + 551626225034158621029168237408 )x^{6} + (-3688847609734369205680134920a + 59778021634718282871961053456 )x^{5} + (-272321209225757099693542589744a + 134180390926147924322645471480 )x^{4} + 189445367243258342154185809584a - 166140617175296818336022802560 x^{3} + (-574719526053856140396965532416a + 479034307073713172046207122840 )x^{2} + (-75864623679389948216653286784a + 320345872458903330842889833008 )x + 430435480205734572541439054688a - 367568235954783932791341829562 \)
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