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ex.24.10.1.127_255_383.a

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

Inertia Polynomial

The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + -351750906685552534296702778256a - 589153484449285808298340694984 x^{47} + (4993859371827445398914680364a + 318666066033493609253555196352 )x^{46} + (-519262284410474239637307800996a + 427465688157445612297731901440 )x^{45} + -526765432932206752777376360756a - 319450071258289444675161428556 x^{44} + (298010432545987623410545400576a + 539498501307255641210899200224 )x^{43} + (9037501494035767700710946676a + 16013294762854880693224258236 )x^{42} + 359592047343434644070334387000a - 96235966275471331437227586840 x^{41} + (-299145958442317173402587905000a + 55418340594106023026193353908 )x^{40} + (-183411727911515051222024487080a + 596464376061297085871485963856 )x^{39} + 403915557810897707617055463852a - 146366952987958450487630703232 x^{38} + -356495696408424112080143212940a - 191619962825702341401302403952 x^{37} + (-400142543541473000424477405552a + 574122294597949410801731544380 )x^{36} + (414298666635876957146863768936a + 258068465423361378079283029936 )x^{35} + (-312199583874231005682962520724a + 324461232858662582377629986864 )x^{34} + (-477153325803879162611644282728a + 389590205898741052362467469160 )x^{33} + (157986618358302412359156355152a + 96165461579204151087185235480 )x^{32} + 542751011286315435952704294816a - 278709655847737320339495685952 x^{31} + (435106455691244727471262159000a + 341655989921097726853413989232 )x^{30} + (414734177251388581688941782568a + 426317443553109020504807874536 )x^{29} + (593207207045237859014170569360a + 412973703162435923985323468736 )x^{28} + 463636583715086042524452476912a - 250953530773712970466246561376 x^{27} + (552940764794443210779671761828a + 155082431977859879668080681752 )x^{26} + (-615835303668827293502740960080a + 537367222541499138430187829872 )x^{25} + -207323331292882063899199076370a - 460932146663736075096855600592 x^{24} + 111769280546136138126860711016a - 518410346059600804137784443616 x^{23} + (576489882718539075472518132296a + 491870370782731519639954390104 )x^{22} + (-243726782665771390680235895280a + 340895362182860605637443856472 )x^{21} + -456409378576261501904934409364a - 264054978109440362802160431320 x^{20} + 459657917849176702867730901328a - 355452648423636261606090612016 x^{19} + (519689929853009345617527199196a + 369897147725736581707597411880 )x^{18} + -323554918221042223665520393432a - 247623755690119325918659028160 x^{17} + 2420621004440505533312406964a - 371627989601175074183201881792 x^{16} + -534672694837530634774380722608a - 151557055089464437541482316160 x^{15} + -179898438011613957630110793528a - 221260774502197672277241962920 x^{14} + (150472523021541418733329412880a + 138146653407817579810805916968 )x^{13} + -239873606322811012071845770272a - 1603788334621795276236430616 x^{12} + -598262074237100199293368252768a - 27459010566908807184304880304 x^{11} + -505361836790857477161431040648a - 255001001066752647028852128680 x^{10} + -524163171520828507015584942008a - 446243909389411068809475156656 x^{9} + (-428127110396412966316522145224a + 275445731365804620788212662000 )x^{8} + -376415515790173558296535211712a - 548508574563764298799525716784 x^{7} + -499563278318768874860235198544a - 519574067030151606208746464000 x^{6} + (-223903001479240963760952036312a + 628491436766818351419709240112 )x^{5} + -500453462558052634075343217408a - 144142319076638503112196452608 x^{4} + 428288048449992021456154028048a - 574284494169461868877259555136 x^{3} + 375366440598193365217417271768a - 559345208915865870117799590616 x^{2} + 274327031089117984778776449632a - 48068965082114645874775488960 x + 234874116478999693283084809064a + 391699184853180379284178089494 \)
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