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ex.24.10.1.127_255_383.c

Base Field
\(F = \) 2.1.2.3a1.3 \( = \mathbb{Q}_{ 2 }(a) = \mathbb{Q}_{ 2 }[x] / (x^{2} + 7\cdot 2 )\) 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 \(x^{2} + 90546471444873528678335943241b^{4} x + b \)

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

Inertia Polynomial

The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + (-541732284167565660751361729168a + 156839231132951266892244744136 )x^{47} + -309291325866762739723700828816a - 409583490278649329391407632316 x^{46} + (490860977358674177651508951228a + 379002319934989554742042448880 )x^{45} + -149704117949215764773177320200a - 145263247480330097677097434336 x^{44} + 244324197902265462704368632640a - 177397904024123579742981123488 x^{43} + -206950366391205187849181295028a - 76947788727981960163384020644 x^{42} + 290799672702115720347145142928a - 394755079100481700289340297960 x^{41} + (32185782059819295024790569056a + 518503047522734268914895356060 )x^{40} + (-503535678269498421659011013816a + 142354738818200595937127623392 )x^{39} + 95994415783576963827799535168a - 34808225425669518381585786936 x^{38} + (-296560593980596451673688642628a + 304101541143028211481793176064 )x^{37} + -133332486553624555128760096800a - 572041946462552199199771220004 x^{36} + (-515609080496576924211350585000a + 583865610948059489377331284992 )x^{35} + (-484019632003101409353110676592a + 469967835111615160160178847976 )x^{34} + 154372395000790843781008955176a - 131969104985636976838575102096 x^{33} + 377670100499100059612442636000a - 302582946699464310063861277984 x^{32} + -398102271881031952628040821824a - 449088906986938766327058602240 x^{31} + (127274362559266992362323102936a + 458413672037648838587436942608 )x^{30} + (-198213551840632227725138319096a + 529730477118452531266862926080 )x^{29} + 438613151054876660037197781836a - 606105793358320582780157247496 x^{28} + 47699698079914627017273594160a - 321948816384408277309503969936 x^{27} + 220206398423014946146080189544a - 372354447406160867388018591800 x^{26} + (-496642645791272283475919486632a + 18878048288578883432852551704 )x^{25} + (-484995348978945786087354983210a + 258345056556184442925412110800 )x^{24} + (289969449381302538731089285944a + 104298029746915383777405240928 )x^{23} + (-426935628441911743620697635148a + 541918347110768189222532724864 )x^{22} + 480496291341619769773541880256a - 94541947662449821943651385688 x^{21} + (287306953788344938274508102672a + 186470814338976412882797444128 )x^{20} + 36610467546370602613964495616a - 271663063626516815472565381776 x^{19} + 606601035290061276730012944188a - 239897707441193432369180416792 x^{18} + (-371340199304382587226103475256a + 381851334224201636001113544224 )x^{17} + -434287588643102687902503476284a - 586787104991277609940643883888 x^{16} + (384332244532401345686045072480a + 629320646838493315450199903008 )x^{15} + (-427088593547633382588862624632a + 331120714725664215319990495424 )x^{14} + (631452490667676280033298648816a + 184448661388503307950804016872 )x^{13} + -518235537200487557126183221016a - 37470776502265832656567391608 x^{12} + (274984079277189695875282168592a + 132568907945216145276634252816 )x^{11} + (33070735091587139220286019504a + 378193869705690815482413988400 )x^{10} + -150955013896223397329213388704a - 72070173758629953160909001744 x^{9} + (370979633121499291877445826832a + 205889915995097299788671163584 )x^{8} + (-608899940847415963324544838496a + 132197251498971495184873396272 )x^{7} + (-129686205120143310945304920976a + 87889169057439726476408247648 )x^{6} + 202783612352246448185977749632a - 85353975387484670765331486192 x^{5} + (280559307378836397599920012552a + 442078945713498920989741402920 )x^{4} + -175860998512523475028112886896a - 347244543494916379714604101248 x^{3} + 518072082753956757298556192584a - 540171946674716966237868340496 x^{2} + 380267505883422043281242104376a - 70910737411230975456927178224 x + 398955690602265049554142101808a - 597464014053465673754405453654 \)
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