← Back to 2.1.2.2a1.1

ex.24.8.1.31_63_95.a

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
\( v(N) = 8\)
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 \(x^{2} + ((a + 1)b^{3} )x + (-\mu_3 - 1)b \)

Underlying Character

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

Order
4
Conductor exponent
14
Values on generators of \((\mathcal{O}_K/\mathfrak p^{ 14 })^\times/U_{\mathfrak{p}^{ 14 } }\) :
\(\begin{array}{l} \chi^A\left(\mu_3c + 1 \right) &= i^{ 1 } \\ \chi^A\left(((2a - 2)b^{2} + (-2\mu_3 - 2)b + ((4a + 2)\mu_3 + 4))c + ((a + 1)\mu_3 + (2a + 3))b^{2} + ((a + 1)\mu_3 + (2a - 2))b - 2a\cdot \mu_3 + a + 1 \right) &= i^{ 2 } \\ \chi^A\left((((a - 2)\mu_3 + 3a)\cdot b^{2} + (4\mu_3 + 2a)\cdot b + 4a\cdot \mu_3)c + ((2a - 2)\mu_3 + (2a + 2))b^{2} + 4b - a\cdot \mu_3 - a + 1 \right) &= i^{ 0 } \\ \chi^A\left(((\mu_3 + 1)b + (2a + 2))c + ((3a - 2)\mu_3 + (3a - 2))b + 1 \right) &= i^{ 0 } \\ \chi^A\left((\mu_3b^{2} + ((2a - 2)\mu_3 + (2a - 2))b + 4)c + (2a + 4)\mu_3b^{2} + (2a\cdot \mu_3 + 2a)\cdot b + 1 \right) &= i^{ 0 } \\ \chi^A\left((-b + (2a + 2)\mu_3)c + (a + 2)b + 1 \right) &= i^{ 0 } \\ \chi^A\left((((2a + 4)\mu_3 + 2a)\cdot b^{2} + 2a\cdot b + (4\mu_3 + (4a + 4)))c + (2a + 2)\mu_3b^{2} + (3\mu_3 + (2a + 4))b + (-2a + 2)\mu_3 - 2a - 1 \right) &= i^{ 2 } \\ \chi^A\left((((a - 2)\mu_3 + (2a + 2))b^{2} + ((3a + 4)\mu_3 + 3a)\cdot b + (3a + 2))c + ((a + 3)\mu_3 + (3a - 3))b^{2} + ((3a + 3)\mu_3 - 2)b + (4a + 2)\mu_3 + a + 3 \right) &= i^{ 2 } \\ \chi^A\left(((3a\cdot \mu_3 + 3a)\cdot b^{2} + (2a + 4)b + (-2a + 2)\mu_3)c + (2\mu_3 + 2)b^{2} - 3b - \mu_3 \right) &= i^{ 0 } \\ \chi^A\left((((2a + 4)\mu_3 + 2a)\cdot b^{2} + ((2a + 4)\mu_3 + (3a + 4))b + (3a + 4)\mu_3)c + 4b^{2} + (4\mu_3 - 1)b + (2a + 1)\mu_3 \right) &= i^{ 0 } \\ \chi^A\left(((3a\cdot \mu_3 + (a + 2))b^{2} + ((a - 2)\mu_3 + 2a)\cdot b + ((a + 2)\mu_3 + (3a + 2)))c + ((2a + 2)\mu_3 + (a - 3))b^{2} + ((a + 1)\mu_3 + 2a)\cdot b + (a - 3)\mu_3 - 3a - 3 \right) &= i^{ 2 } \\ \chi^A\left((-\mu_3 + 3)c + -\mu_3 + 3 \right) &= i^{ 0 } \\ \chi^A\left(((a\cdot \mu_3 + (a + 4))b^{2} + ((2a + 4)\mu_3 + (3a + 4))b + 2a\cdot \mu_3)c + (4\mu_3 + 4)b^{2} + (4\mu_3 + 1)b + (2a + 3)\mu_3 + 4a \right) &= i^{ 0 } \\ \chi^A\left(((-\mu_3 + 3)b^{2} + (3a\cdot \mu_3 + (a - 2))b + (4a\cdot \mu_3 - a + 4))c + (-3\mu_3 - 3)b^{2} + ((a - 2)\mu_3 + (2a - 2))b + (2a - 2)\mu_3 + 4a + 1 \right) &= i^{ 1 } \end{array} \)

Inertia Polynomial

The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + (-265082562469940717161261658044a + 272816921959766874900991036952 )x^{47} + (469011253285532034665899600128a + 585631248043327827146506968824 )x^{46} + (-332380906456710770183198092924a + 194395158110630915642758385376 )x^{45} + 596057523278329544660438765500a - 273951461186744748017414139664 x^{44} + 312484274434005594446662614496a - 432593495543281272258396344656 x^{43} + 196187717853789187712392595724a - 212856752309431596379057857292 x^{42} + (-3339644022986921433148762840a + 561928343740498100963217579228 )x^{41} + 73386243196006510582413309208a - 531218561152907697453197365932 x^{40} + 584496676839926083813626784104a - 90792148852180240350271643040 x^{39} + (-381603788221983150609593803776a + 128175468168583615972742513172 )x^{38} + 305936941457250424353165313340a - 64271125904786368782015649044 x^{37} + (176287451907696349716145319576a + 104565006147210486524827171756 )x^{36} + 25856378791101915730818681536a - 588219597882899958357822907576 x^{35} + -385170000144509536893452016416a - 399861789752535471888646643296 x^{34} + (154018458267288026663896856248a + 358161977467798007496546037088 )x^{33} + -310391458154967685486792834892a - 70339189164820308064291374756 x^{32} + -268354686783694796495468230912a - 101380895302785814408599197388 x^{31} + (-316883775950616961387851834764a + 267321397610400669051917827044 )x^{30} + 515622646427709069872144223068a - 146811225385245051581543222704 x^{29} + (2698657483043952156147871516a + 373873827647903335547744054016 )x^{28} + (-222176008928238113066297242240a + 136474942188018356588976484080 )x^{27} + (564366754792985477381552700664a + 623383447017981601697637750960 )x^{26} + (-370219306965297755501022589288a + 390714387459351264108075876192 )x^{25} + -477560140356822703745989546858a - 7728807763291767599026088952 x^{24} + -192716758727052534098365243952a - 424370142213336075860608127616 x^{23} + 49597762743528643001570562920a - 337479274850658461716750327240 x^{22} + (-128077624508242911301311859608a + 520589532526878283232806585784 )x^{21} + 143046147414880753816926855256a - 141655036286199252456739398584 x^{20} + (-426204371123996848325745723360a + 196669534310525124876368629968 )x^{19} + (254796171404902479942967007076a + 12438936863935774019272236080 )x^{18} + (-573374775081504772121225629020a + 37828584868453985621498982648 )x^{17} + (473842205492094474460385284748a + 96052411278220467961137311736 )x^{16} + 573844591135191081492228880880a - 257003038969423740887566620400 x^{15} + (-297824669853012624004713271852a + 460881200268925271442275221344 )x^{14} + -71353084039850327039921405500a - 342078419205056630566633471048 x^{13} + (94035237372435717317802102596a + 267906872946244550867179075652 )x^{12} + (19878132179075550900213891912a + 529425141163371970100934073424 )x^{11} + -283689298783166102071806473984a - 182702928292968514997759544240 x^{10} + 579654679555358070696748247200a - 517380961203275334282157602144 x^{9} + 614380960872952613473143898396a - 353814780891840476462717953936 x^{8} + 136483109055478203246353836132a - 310772821216545370250369948840 x^{7} + (-142069153516685036382472236236a + 469595876359217187905256588328 )x^{6} + (480470439862013982333696648968a + 461487161610190147725152439864 )x^{5} + -603958206704877471491301728440a - 474668432598768239444767119720 x^{4} + -164035433855632696307754401040a - 452387837212833503191585149008 x^{3} + (-233197285604606048623703995624a + 336998908373783041736532780496 )x^{2} + (-245018431634900047449641180264a + 606628146251053856990922901680 )x + 275253308318132987342764262802a - 524306774182389714941158072262 \)
← Back to 2.1.2.2a1.1 Summary