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ex.24.10.1.131_259_387.d

Base Field
\(F = \) 2.1.2.3a1.4 \( = \mathbb{Q}_{ 2 }(a) = \mathbb{Q}_{ 2 }[x] / (x^{2} + 3\cdot 2 )\) 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 \(x^{2} - 211275100038038233582783867563b^{4} x + 3b \)

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} + -339930483393818779143136130224a - 592799869674995720243091701464 x^{47} + (571269350765906033137783688332a + 509783746558795468949756773780 )x^{46} + 70199683128068813707885767332a - 131865737454749991609746179344 x^{45} + -251963093146500272224905007412a - 200758948282292961837846869072 x^{44} + -483514125059439041346023810808a - 399326963838349314942126449792 x^{43} + (-109168036461744320346249964092a + 305585631733650335230200645660 )x^{42} + -275870343893622628893239487944a - 156622226124362382721068707272 x^{41} + (-364400202290326878786721422152a + 586904754269286413356742628148 )x^{40} + 12200518194810651906600546216a - 42809350152588150245776311728 x^{39} + (-357308511941045046633254848600a + 327213762044204119133457780320 )x^{38} + (-105009386836313011273847455060a + 194152202260808905767146170128 )x^{37} + -254442418499046502742166676688a - 146402801030076962229759729148 x^{36} + (569193532168252967215015522400a + 388887104433423842927553452704 )x^{35} + (-154993759443793162353748502836a + 114552797454266423806037145376 )x^{34} + (74334126402972470651620255416a + 627526918188191548310891209920 )x^{33} + (-440557475368790427389950694748a + 66128772851407530459453099520 )x^{32} + (-261020434779766097748735687488a + 174696714206288805294056136224 )x^{31} + -552643575609050154280673878120a - 399373722118688134396593088176 x^{30} + 258188054951607867240918972144a - 394815220812341644782847999696 x^{29} + 262935368739275882894677716a - 286851196901977298459053690968 x^{28} + (-594498561198947706732491172816a + 91001163820350430252586782736 )x^{27} + (68885932892658955397120791624a + 375273048719707407937299589976 )x^{26} + (584274786167165894915172726712a + 162085183581445312709356355120 )x^{25} + (607474682937220097637235880610a + 380233090411714304241988909956 )x^{24} + (-600469268552009415772373002344a + 112979023924798115315073964496 )x^{23} + 111300994094471947699866252116a - 467284294861094679104199712528 x^{22} + 117863771533965451488510949624a - 319803612277801628796434890840 x^{21} + -229664282718978635544382965128a - 85577363895110065733771735512 x^{20} + (489418465928379763424252039424a + 40172961843292992351157771888 )x^{19} + (-118967107933698851208033682532a + 284984370286062517104872504224 )x^{18} + 315532535207389975044614795208a - 180639576535767000235734927648 x^{17} + -335587990054712586550438038156a - 575849881171879096810441137432 x^{16} + -213462461358070975322415728848a - 620661973411405872524637498336 x^{15} + (76235800688418176166749595600a + 487142880861707562302902030864 )x^{14} + 474196418665279711624683537544a - 235378042827011132608629804424 x^{13} + -116199693940893722281612517144a - 83131072164130589291196470472 x^{12} + -456543521120486771305236813328a - 138386560212077741722648745248 x^{11} + -386218715245936363233757706400a - 611233892667906219038183262936 x^{10} + -292158767118308409201390599488a - 254353273749854021077514115696 x^{9} + (-567933432973018568470775243368a + 441368368161569839212316075480 )x^{8} + -552776465827918731534841797248a - 418872791280162358215750007888 x^{7} + -282628488955338095578746880752a - 339446125485146961631345384768 x^{6} + 457073016204504691065703209600a - 94818185222049597125964150144 x^{5} + (244901724159802417247117178880a + 615250735633604211217335854264 )x^{4} + 287071029969855328771585226032a - 583372492680150329929857160928 x^{3} + 243097883002728313909539019432a - 509365860259878462977037737200 x^{2} + (375341861469293421780989420336a + 252533645277644748831164738160 )x + 221177888428773310272790154956a - 615565611932042626388695989274 \)
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