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ex.24.10.1.131_259_387.d

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

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

Inertia Polynomial

The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + (179899114500228288783504819408a + 380101234048457370666818123832 )x^{47} + (-323017092494223668527422953972a + 352411375739801237324761949344 )x^{46} + (21744789993159061343495563812a + 623694465601258470420076191184 )x^{45} + (-564748746625564408472796826456a + 370275661955552862003556935188 )x^{44} + 333797550124250635779607808712a - 315099556851423332786344278608 x^{43} + (-591388781177299167767857892716a + 442584037629009488254415673692 )x^{42} + (-413543216025704911159755259904a + 381729687762306981297273075056 )x^{41} + (513064579536371940351488726644a + 506182665652360124732675092364 )x^{40} + (159227823612928681760216591416a + 117971673026681506537497214160 )x^{39} + (172772351453999591015618079212a + 533389778039783220281571239832 )x^{38} + (299887000149234264453566033364a + 626892882698074766859196063640 )x^{37} + (-344094586797949226300829585732a + 332660710915725835850946069900 )x^{36} + (385966831039941516833347617648a + 167519994901149366590704114080 )x^{35} + 98508412484998973691557020580a - 560549229608458059101200507888 x^{34} + (-578661392269421406957087654632a + 262553567368121416728992595424 )x^{33} + 186167249666169567676732618016a - 36576467761546837059314752832 x^{32} + 560898283180418582270665293440a - 145211071505472319753319023168 x^{31} + (-290924842075971903809027297760a + 616057079610554018971603829536 )x^{30} + -584544301021042386564454169312a - 366751146436606548513742003624 x^{29} + (-114176920329378267354740400008a + 39905060764251964430263342184 )x^{28} + (-299784900468257916012454240912a + 136257216732401995912534990032 )x^{27} + -510549733801457361433131645460a - 274064015138242695733219173424 x^{26} + -208437004860410090449569815024a - 254083335344776805905209377192 x^{25} + 23958695961582266844607700282a - 632130306813849888639058062036 x^{24} + (234376484940091583213321107688a + 457422002853777219914755917040 )x^{23} + 236857820219225400648081942032a - 222250777128060271396294260536 x^{22} + -86550342468768341489925570744a - 423182858313726414018761446888 x^{21} + (-509935313767902839476775252460a + 343859062551415534854779791304 )x^{20} + 298020703545948669982431481968a - 170883956339597968154515962352 x^{19} + 36279792072739241397500642428a - 138727113586039185965056567168 x^{18} + 71726355685028457936651311376a - 219872353221436610964328996656 x^{17} + 406025822121376625954193627452a - 81638818024621134075792221984 x^{16} + 166335727544201050791639353776a - 383288520616802621670641790784 x^{15} + (-107002806027523377387208904136a + 344296768033457324302385327368 )x^{14} + (-93145444109682816405489288912a + 297267281532830788221193274184 )x^{13} + 497419943653049883865266381504a - 85981815075539503217381475920 x^{12} + (542397610888851943009661458640a + 553115507020868382471998126688 )x^{11} + (-9124555703315236559028573104a + 312400536323529631611475706920 )x^{10} + (577701786818947176872146270288a + 271167517882104346228606125744 )x^{9} + (-325244294255006684364002888032a + 389291390284711771061597714320 )x^{8} + -578917671091099687251328032224a - 418019018986215387503725205936 x^{7} + (-301526278444362745987043779792a + 206836977800649376673774867168 )x^{6} + 64735297572997546065977888536a - 262326205937574963869624636784 x^{5} + 409719073280392154868084519240a - 305197379429217277751461715712 x^{4} + (-21740755098165115159946491696a + 533308864693721845565661800096 )x^{3} + (258391910553665164240810375592a + 363988607363341463856603572392 )x^{2} + (-99525660275259167722676288120a + 599645011357356999629503333328 )x + 483578642255775189597130081220a + 89971195413744119369423776634 \)
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