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ex.24.10.1.31_63_95.b

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
\( v(N) = 10\)
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 \(x^{2} + ((a + 1)b^{3} )x + (-\mu_3 - 1)b \)

Underlying Character

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

Order
4
Conductor exponent
20
Values on generators of \((\mathcal{O}_K/\mathfrak p^{ 20 })^\times/U_{\mathfrak{p}^{ 20 } }\) :
\(\begin{array}{l} \chi^A\left(\mu_3c + 1 \right) &= i^{ 1 } \\ \chi^A\left(((2a - 2)b^{2} + (-2\mu_3 - 2)b + ((4a + 2)\mu_3 + 4))c + ((a + 1)\mu_3 + (2a + 3))b^{2} + ((a + 1)\mu_3 + (2a - 2))b - 2a\cdot \mu_3 + a + 1 \right) &= i^{ 0 } \\ \chi^A\left((((a - 2)\mu_3 + 3a)\cdot b^{2} + (4\mu_3 + 2a)\cdot b + 4a\cdot \mu_3)c + ((2a - 2)\mu_3 + (2a + 2))b^{2} + 4b - a\cdot \mu_3 - a + 1 \right) &= i^{ 0 } \\ \chi^A\left(((\mu_3 + 1)b + (2a + 2))c + ((3a - 2)\mu_3 + (3a - 2))b + 1 \right) &= i^{ 0 } \\ \chi^A\left((\mu_3b^{2} + ((2a - 2)\mu_3 + (2a - 2))b + 4)c + (2a + 4)\mu_3b^{2} + (2a\cdot \mu_3 + 2a)\cdot b + 1 \right) &= i^{ 0 } \\ \chi^A\left((-b + (2a + 2)\mu_3)c + (a + 2)b + 1 \right) &= i^{ 2 } \\ \chi^A\left((((2a + 4)\mu_3 + 2a)\cdot b^{2} + 2a\cdot b + (4\mu_3 + (4a + 4)))c + (2a + 2)\mu_3b^{2} + (3\mu_3 + (2a + 4))b + (-2a + 2)\mu_3 - 2a - 1 \right) &= i^{ 2 } \\ \chi^A\left((((a - 2)\mu_3 + (2a + 2))b^{2} + ((3a + 4)\mu_3 + 3a)\cdot b + (3a + 2))c + ((a + 3)\mu_3 + (3a - 3))b^{2} + ((3a + 3)\mu_3 - 2)b + (4a + 2)\mu_3 + a + 3 \right) &= i^{ 2 } \\ \chi^A\left(((3a\cdot \mu_3 + 3a)\cdot b^{2} + (2a + 4)b + (-2a + 2)\mu_3)c + (2\mu_3 + 2)b^{2} - 3b - \mu_3 \right) &= i^{ 0 } \\ \chi^A\left((((2a + 4)\mu_3 + 2a)\cdot b^{2} + ((2a + 4)\mu_3 + (3a + 4))b + (3a + 4)\mu_3)c + 4b^{2} + (4\mu_3 - 1)b + (2a + 1)\mu_3 \right) &= i^{ 0 } \\ \chi^A\left(((3a\cdot \mu_3 + (a + 2))b^{2} + ((a - 2)\mu_3 + 2a)\cdot b + ((a + 2)\mu_3 + (3a + 2)))c + ((2a + 2)\mu_3 + (a - 3))b^{2} + ((a + 1)\mu_3 + 2a)\cdot b + (a - 3)\mu_3 - 3a - 3 \right) &= i^{ 2 } \\ \chi^A\left((-\mu_3 + 3)c + -\mu_3 + 3 \right) &= i^{ 0 } \\ \chi^A\left(((a\cdot \mu_3 + (a + 4))b^{2} + ((2a + 4)\mu_3 + (3a + 4))b + 2a\cdot \mu_3)c + (4\mu_3 + 4)b^{2} + (4\mu_3 + 1)b + (2a + 3)\mu_3 + 4a \right) &= i^{ 0 } \\ \chi^A\left(((-\mu_3 + 3)b^{2} + (3a\cdot \mu_3 + (a - 2))b + (4a\cdot \mu_3 - a + 4))c + (-3\mu_3 - 3)b^{2} + ((a - 2)\mu_3 + (2a - 2))b + (2a - 2)\mu_3 + 4a + 1 \right) &= i^{ 3 } \end{array} \)

Inertia Polynomial

The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + (-239840388203287867108183634236a + 382068097111772142041242661776 )x^{47} + 513505779349754298001212401260a - 69966361862378417240930405608 x^{46} + (-114015407512874769869124786432a + 37963583820891308487277755008 )x^{45} + 453076170967078549980523942336a - 320324835407139077757737257808 x^{44} + (-379345754598631590020510475640a + 115000846290293564707916581692 )x^{43} + (-43610064126744469813733158596a + 106470658270606193033101200056 )x^{42} + 33289228271556963776127854092a - 276448755638852806512616413984 x^{41} + (-402088404574194025052949641440a + 465933016886382915697357607264 )x^{40} + (-136948231586603199183855915144a + 479407570926131128680345209248 )x^{39} + (-480798848453323681459902960264a + 73937702666773161843855467884 )x^{38} + 344998192222763705003772070968a - 493910491316284177313653549680 x^{37} + 504005531131280693437918769284a - 383293851439541248320419038784 x^{36} + 596329635452512610139627141744a - 392327059301707614721764386736 x^{35} + 359090110502173404272171112216a - 548716998267439666349734347824 x^{34} + -590829634682740233639898842808a - 613414600111460400430202462040 x^{33} + (278719430066038857133967043328a + 271197141075724654811460854504 )x^{32} + (-569960425190941210824366178420a + 532734037947744382698392214576 )x^{31} + -472334009070044687142996040028a - 327266658050004555337212602324 x^{30} + -190629203306873708575778696972a - 548670346938937652204808315600 x^{29} + (-175000632111221058132056547144a + 179635338516603938765452258696 )x^{28} + (-129642464547412897790232296672a + 18900956251178956787926430752 )x^{27} + (-212271983426205713782480629432a + 75599697325846429785601030248 )x^{26} + 572123926369605918590490674112a - 147263772618774779229851813608 x^{25} + -281965939892687109235115170122a - 408880746298693590325048984944 x^{24} + (-530913179275146077912780777888a + 386631217866763739404748603368 )x^{23} + -586952082935457342705137374384a - 521751183423797270347281103336 x^{22} + -222609921968929792605946246752a - 184315907280436970892478378016 x^{21} + 535294203587312234256342491648a - 591798361640282539677089457112 x^{20} + 228045288711689785405462899748a - 11873436663342174394289160624 x^{19} + (181916366132592336374238827064a + 554506049600965100710086761944 )x^{18} + (-345364172171954080473586929288a + 189553773787128560072409485128 )x^{17} + (483973138065560250239357452448a + 139357035679093313620404367856 )x^{16} + -431766603937023529580410805488a - 429552932008212260461573691680 x^{15} + 144082433312320587265866599924a - 463174269718798209764336876912 x^{14} + 339506634996711908208866954328a - 344745835320118695539053674816 x^{13} + (10608868522306045615952019288a + 595246624932801878470668731044 )x^{12} + 144023541659537151006014156040a - 459202420959253416598210334384 x^{11} + -40743811163038102570956580384a - 103928594802241103461619156432 x^{10} + -289490289544920881422186200600a - 286866091589510601634982770928 x^{9} + 507416939314953618022976353640a - 331694108454464778588562722960 x^{8} + (-538580587574386595475304221464a + 83169621663809919496528289384 )x^{7} + (535633929584872626670677195972a + 219775400988562029748619300344 )x^{6} + -201493780966049821354475739240a - 87211070652946359718388948696 x^{5} + 311104244297851780261141831480a - 51335066755410946947934455680 x^{4} + (-378777456551361135306021351168a + 396572945292826818247906021888 )x^{3} + 491654469142556534930901684728a - 192494989079967994938745912224 x^{2} + -51040520734714061008606432976a - 175782879191675694356673863344 x + 295705634174705516576556451590a + 206416652627224970057549614782 \)
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