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ex.24.10.1.127_255_383.c

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 \(x^{2} - 211275100038038233582783867563b^{4} x + b \)

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

Inertia Polynomial

The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + -294752229081738125607234122768a - 528931595222258154223109781144 x^{47} + 372544862585387674978168164848a - 127216267802861460859466936380 x^{46} + -316154333050234385391542478276a - 429320535230301762614067199120 x^{45} + -76085209350448600673621019464a - 621458606147821182528657656272 x^{44} + (231427203863591072087166517280a + 70195480021233159214411956640 )x^{43} + (265936191115105833904996628636a + 470830169310095181076694521580 )x^{42} + (-218910529978430541052619007584a + 480497576055368207359804500376 )x^{41} + (-128567951770122654185550737168a + 152212027643377383559560281964 )x^{40} + -463070701789411191462627403288a - 450161550168822681926431166112 x^{39} + 592864365639953858876437519440a - 353439533117276646267028585320 x^{38} + (-579859777293545950683780430420a + 11723992750625662193979645408 )x^{37} + (392076423044428013552103018064a + 327570742926242815578706906012 )x^{36} + (394030327179959194523634336440a + 455978478695199934855850290816 )x^{35} + 523613202188218572044239430336a - 561787565851244752990339864984 x^{34} + (223861363861706783315613860072a + 306264823826297653041892646768 )x^{33} + (-114398998577626324955568617424a + 602950984377046776821517165232 )x^{32} + 79265573151865243104268636352a - 248108208294892226152089413248 x^{31} + (-184773182512264074062263643368a + 488047223260277551059424983888 )x^{30} + (286970005940933447030175972952a + 142517884077777982264018351744 )x^{29} + 360483034872623300543642491004a - 616748556061701159970948778120 x^{28} + -42212678130373841284823865552a - 451173637265032768188925521808 x^{27} + 569642980224578083292607099576a - 269689052541926049046983923368 x^{26} + (506812200376925506628574410456a + 393041591158368157781707154488 )x^{25} + (-9284440758718005731304931562a + 383823899984317900368417818064 )x^{24} + (379268702938374825828690548760a + 233385553559443339544044391008 )x^{23} + -393248801450972972445350073548a - 105166413537855163513324726272 x^{22} + -193130209512000261725056502144a - 99338984772410485525013653176 x^{21} + (-41505339652505098147333304032a + 373268890343794034767244149952 )x^{20} + (-105888823372496851388254680640a + 367563417613052452071831130736 )x^{19} + (-496247684485459121889289312692a + 429673533131943352851703985192 )x^{18} + 125198858806204880921051903560a - 246892438695993945604002515520 x^{17} + 602709596520704431129056424724a - 66929704079021561139857677968 x^{16} + (317750621972841423303142345248a + 135884825328056025307231516704 )x^{15} + (616064233979064714509479659672a + 331820351249759572613846812384 )x^{14} + -178485295047949108466478200880a - 510307085240270849303818998040 x^{13} + 224765112824947144398513480392a - 190706084985875124024924277816 x^{12} + (-350472434797881619324854173232a + 443366461023053412633002854672 )x^{11} + (-609254518886106940252355662288a + 495551443941398962895968010960 )x^{10} + (308890377409431598612068726880a + 49747507225336509339650590128 )x^{9} + -24677011562340004740451006496a - 356650660261920182486298540256 x^{8} + 607631745486353156440143150496a - 220329829857735605761675547920 x^{7} + 382299813063087226772022509936a - 323127605105981462626672046496 x^{6} + -115528349921292302527822762112a - 558418845002919858251600535888 x^{5} + (-237371475287420697120500901048a + 550350801315841258527089731816 )x^{4} + (-167826177815811685817864151152a + 503619803322287726205774228608 )x^{3} + 437996956618198922370043272632a - 385962935947102220161687881072 x^{2} + -481572352431817657802577076328a - 73300775322094536579414403248 x - 289726784414451239016089939088a - 35133386139693090342020645486 \)
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