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ex.24.10.1.131_259_387.d

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 \(x^{2} + 90546471444873528678335943241b^{4} x + 3b \)

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

Inertia Polynomial

The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + -248992593236049286289552012336a - 547095474376502498932031775608 x^{47} + -344748938464021948977804888820a - 441835087538208029193815472748 x^{46} + (134690713373349408416302071076a + 600888819147721634750146867056 )x^{45} + (-222901301912741031548581774788a + 331716356120903642505394015520 )x^{44} + 506544126056963050309792612008a - 99490359071860308401603960064 x^{43} + (339610314789784209174016206836a + 558061981752698996979241528844 )x^{42} + 546515676012872940306188213512a - 612665340722043476822754442440 x^{41} + -516065060016772757528054819384a - 487605388114743239864322887164 x^{40} + 425559934264571489887222955464a - 189545978754436409095586471728 x^{39} + -631099459890639683136648403496a - 228762994279919913045224815152 x^{38} + 85975262007331639427551345276a - 367171985328832584160282734928 x^{37} + (-327696577294236690243296151872a + 250419268545635122708241659684 )x^{36} + 420656500916838185524554433536a - 321743533308747167147266878496 x^{35} + 4066957278060377155765038780a - 191366784294805398146733591104 x^{34} + (-400350740921425310972241186888a + 556564061625491651833148313984 )x^{33} + (478267840687823215180626687348a + 298002342785382853439969739008 )x^{32} + (353597116890824366865143537472a + 293262257528267101247291150240 )x^{31} + 302079480385005683727836379992a - 234190287152368462333628024880 x^{30} + -598124134227982032372971795520a - 391998046012131752022498348560 x^{29} + (-5779170931291667509815872060a + 309705504039367720080340039816 )x^{28} + -455241842471635438342076232400a - 216073247491964423154128244464 x^{27} + -14435130005965543613354933064a - 334610261969356159466739230456 x^{26} + -339688640601631008665127651496a - 353656558492803470001957886640 x^{25} + (583377042602555031205790637426a + 330407903127236088383086030900 )x^{24} + (578465371769442132323765988344a + 470570372485902991849951550288 )x^{23} + (323321239238108078944807596116a + 305315874276349509827896554800 )x^{22} + (-515133199371720113330253534952a + 311203465145956558934041823944 )x^{21} + (-504758261996977927173579910232a + 423215147753865833776278199272 )x^{20} + -336177097771437112317995086784a - 228787946883961658193143002768 x^{19} + 631117354505258777881859088588a - 614137770598936211905189247264 x^{18} + (-296184945519651616519984101720a + 511035113048748023057342346048 )x^{17} + 512401571324240049122052852452a - 265326903223865619050062450552 x^{16} + (338200519274069072515887212272a + 486697307293019054228427920928 )x^{15} + (-176325591619430534147935322432a + 359506544584703583507252422160 )x^{14} + 448425938774856265808962951912a - 593682126847503524079063502600 x^{13} + (-275867249571696854928714479640a + 549361431324845024871790487032 )x^{12} + (-141290791046435606187465315280a + 80675695695220106011068769440 )x^{11} + (119517573396953386033864103744a + 325338962485232930834938189288 )x^{10} + 489197677197063243640313098304a - 534216348646432811888955132336 x^{9} + -594733451938976065366179483432a - 443638662445850214245304458920 x^{8} + -406184886556904163024936925824a - 258783770873055826652116138512 x^{7} + (-580121364015980733431233807216a + 139474077248801760845797063104 )x^{6} + 420043917972916231686043034976a - 322594152394967488356425670048 x^{5} + (455088592769694503638427516512a + 69505523891328764011811584888 )x^{4} + -228407251803463935608275915600a - 33313021727861975280169904352 x^{3} + (-191345232068577522264834318440a + 529975116532285392302971572688 )x^{2} + -325920136865545324957870461232a - 237625996315653375717694401232 x - 459725043930492125655363984196a + 330757608577068529062020039678 \)
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