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ex.24.10.1.33_67_101.c

Base Field
\(F = \) 2.1.2.2a1.1 \( = \mathbb{Q}_{ 2 }(a) = \mathbb{Q}_{ 2 }[x] / (x^{2} + 2 x + 2 )\) 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 \(x^{2} + ((a + 1)b^{3} )x + ((a + 2)b - 1)b \)

Underlying Character

Character \(\chi^A:\mathcal O_K^\times \to \mathbb C^\times\) with the following properties:

Order
4
Conductor exponent
20
Values on generators of \((\mathcal{O}_K/\mathfrak p^{ 20 })^\times/U_{\mathfrak{p}^{ 20 } }\) :
\(\begin{array}{l} \chi^A\left(\mu_3c + 1 \right) &= i^{ 2 } \\ \chi^A\left((2a\cdot \mu_3b^{2} + ((2a - 2)\mu_3 + (2a + 4))b + 4a\cdot \mu_3)c + 4\mu_3b^{2} + ((2a + 2)\mu_3 - 2)b - 3a\cdot \mu_3 + a + 1 \right) &= i^{ 0 } \\ \chi^A\left(((3a\cdot \mu_3 + (2a - 2))b^{2} + ((2a + 4)\mu_3 + 4)b - 2a\cdot \mu_3)c + (-2\mu_3 + 2a)\cdot b^{2} + (2\mu_3 + 2)b + (-3a + 4)\mu_3 + a + 1 \right) &= i^{ 2 } \\ \chi^A\left((((2a + 4)\mu_3 + 1)b^{2} + ((2a + 4)\mu_3 + a)b + (4\mu_3 + (4a + 3)))c + ((2a + 4)\mu_3 + (2a + 1))b^{2} + (4\mu_3 + (3a + 4))b + -2a + 3 \right) &= i^{ 0 } \\ \chi^A\left(((2a + 1)b^{2} + 2b)\cdot c + 2a\cdot b^{2} + 2a\cdot b + 1 \right) &= i^{ 0 } \\ \chi^A\left((((2a + 4)\mu_3 + (3a - 2))b^{2} + (2a - 1)b + ((-2a + 4)\mu_3 - a + 2))c + ((2a - 1)\mu_3 + (2a + 1))b^{2} + ((2a - 2)\mu_3 + 3a)\cdot b + 3\mu_3 + 4a - 3 \right) &= i^{ 0 } \\ \chi^A\left(((2a + 2)\mu_3b^{2} + (3a\cdot \mu_3 + (2a + 4))b + 3a\cdot \mu_3)c + ((3a + 2)\mu_3 + 4)b^{2} + (-3\mu_3 + 4)b + (-2a + 1)\mu_3 \right) &= i^{ 0 } \\ \chi^A\left(((2\mu_3 + (2a + 4))b^{2} + ((2a + 2)\mu_3 + (a - 2))b + (2a\cdot \mu_3 - a + 2))c + ((3a + 4)\mu_3 + (3a + 2))b^{2} + ((2a + 1)\mu_3 + (2a - 3))b + (4a + 2)\mu_3 + 1 \right) &= i^{ 0 } \\ \chi^A\left((((a + 4)\mu_3 + (a + 4))b^{2} + (2\mu_3 + 2)b + ((-3a - 2)\mu_3 - 3a - 2))c + (-3\mu_3 - 3)b^{2} + ((a + 3)\mu_3 + (a + 3))b + (a + 1)\mu_3 + a + 1 \right) &= i^{ 0 } \\ \chi^A\left((((2a + 4)\mu_3 + (2a + 4))b^{2} + (2a\cdot \mu_3 + 2a)\cdot b + (4a\cdot \mu_3 + (4a + 4)))c + ((3a - 2)\mu_3 + (3a + 4))b^{2} + ((2a + 3)\mu_3 - 1)b + (-2a + 2)\mu_3 + 2a + 3 \right) &= i^{ 2 } \\ \chi^A\left(((2a\cdot \mu_3 - 2)b^{2} + (3a\cdot \mu_3 + 2a)\cdot b + ((2a + 4)\mu_3 + (2a + 4)))c + ((a + 3)\mu_3 + (2a + 2))b^{2} + ((a - 1)\mu_3 + (3a - 1))b - 2a\cdot \mu_3 + a + 1 \right) &= i^{ 0 } \\ \chi^A\left(((3a - 2)b^{2} + b + (2a + 2))c + (2a + 2)b^{2} + (3a - 2)b + 1 \right) &= i^{ 0 } \\ \chi^A\left((((3a + 2)\mu_3 + (3a - 2))b^{2} + ((a + 2)\mu_3 + (3a - 2))b + ((-a - 2)\mu_3 - a + 2))c + ((2a - 3)\mu_3 + (2a + 1))b^{2} + ((3a + 3)\mu_3 + (3a - 1))b + (-3a + 1)\mu_3 + a + 1 \right) &= i^{ 0 } \\ \chi^A\left(((2a + 1)\mu_3b^{2} + 2\mu_3b)\cdot c + 2a\cdot \mu_3b^{2} + 2a\cdot \mu_3b + 1 \right) &= i^{ 1 } \end{array} \)

Inertia Polynomial

The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + 439786656758452492784345844464a - 196313674962279585856608394904 x^{47} + (80361096589798503537563125992a + 296769531164423088343434930884 )x^{46} + 416065316849939362663435157180a - 372916168469141548286909361080 x^{45} + -490726560996696183077629047672a - 567887855923372799099741767100 x^{44} + -397040816487492935344371016868a - 76212735400030187120115039612 x^{43} + 489793568119800331818695173580a - 609445781308006530614656497984 x^{42} + (-84279224427401426467426280124a + 127661886977493329299048237032 )x^{41} + 158913807320793982709332832744a - 468906177688517504121169721136 x^{40} + (-117273294509301892315266276112a + 388212392648519876342195844304 )x^{39} + 75515030526908644847047685220a - 484425283543345833873114403340 x^{38} + (459141562222772056033670602368a + 157344566469114005423787948756 )x^{37} + 575456986121342935802728106692a - 198844596397753797365226200664 x^{36} + -339726570591210194041896084520a - 216276866527788273403449577936 x^{35} + -2696410727200578109303489896a - 354981747825940778751258672080 x^{34} + 583527292596783228333870970404a - 534748304675131933023682365000 x^{33} + (-300746380945242563543459359320a + 199549856461687350047847658528 )x^{32} + (622836937276550718690521767252a + 561314841596157548980525114672 )x^{31} + (229351724820337093529341106456a + 525240740347597765623665560188 )x^{30} + (-115207381671199108536290218280a + 77127018107743683757199125112 )x^{29} + 308976535661277081949304107024a - 209184766737163055457260737400 x^{28} + (-64003538617265440907058243928a + 279599621765562218186227062248 )x^{27} + (168890695741745654136107422200a + 372709355715113869903787539732 )x^{26} + -333084261833908980423286506724a - 63365888082587266918630657072 x^{25} + 247412242238650120039448848758a - 574019977605177299297031451084 x^{24} + (-584711260137907231344037962000a + 314395060374394565940528961552 )x^{23} + 558859090809327122519968312684a - 34097769265399969840072796744 x^{22} + (319930966489539324794616443176a + 229576672329566426843235132344 )x^{21} + (99154155595652026128184722764a + 556435533565357861508463538768 )x^{20} + -424047733692891349939541990932a - 370618608992469351077875761320 x^{19} + (-203842713808739568520031713816a + 102396773782967115676625835224 )x^{18} + 600759124974008154081342516720a - 384543360189772783035574693528 x^{17} + (-344284989189107211241368610864a + 631579039326442886363806316240 )x^{16} + (-385849011642669704904496901096a + 365310696375194111297957705024 )x^{15} + (-163428831012075336276962680892a + 449655454708406865454851884208 )x^{14} + (311140287374986456641341294068a + 601049129596952951124855470968 )x^{13} + (-43144252905355726789406625080a + 436381749468813728756544286252 )x^{12} + (-442848698550923554023021929328a + 576488881543814422636662418224 )x^{11} + (581486522291694044159927766448a + 43259911073482887635719047040 )x^{10} + (-245471153841803145287508868312a + 393293434975946145701234653128 )x^{9} + (-61599725869214563030664483344a + 51478598791391238568368855544 )x^{8} + (-585902405149391407239528957336a + 621327597804576385396259813640 )x^{7} + 191595748241413634588544762284a - 531755569650036027781577542568 x^{6} + (-604332935709583793251662766840a + 64509397491767698180633362480 )x^{5} + (-536339065844405818084547569712a + 295408211075357323751030432544 )x^{4} + 90516720152775500256794419576a - 615921878257565492740697027184 x^{3} + -207126806706341077947901857388a - 58692448981988669434614427400 x^{2} + -491894492591014818754152798696a - 73034863723902805537531360440 x - 269483968930817754346760444926a - 4395132138785996910862619182 \)
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