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ex.24.10.1.31_63_95.b

Base Field
\(F = \) 2.1.2.2a1.2 \( = \mathbb{Q}_{ 2 }(a) = \mathbb{Q}_{ 2 }[x] / (x^{2} + 2 x + 3\cdot 2 )\) 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 \(x^{2} + ((70425033346012744527594622521a - 70425033346012744527594622521)b^{3} )x + ((-140850066692025489055189245042a + 140850066692025489055189245041)\mu_3 - 140850066692025489055189245042a + 140850066692025489055189245041)b \)

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 + ((3a + 1)\mu_3 + (2a - 1))b^{2} + ((3a + 3)\mu_3 + (2a + 2))b + 2a\cdot \mu_3 + 3a - 3 \right) &= i^{ 2 } \\ \chi^A\left((((3a - 2)\mu_3 + (a + 4))b^{2} + (4\mu_3 + 2a)\cdot b + 4a\cdot \mu_3)c + ((2a + 2)\mu_3 + (2a - 2))b^{2} + 4b + (a + 4)\mu_3 + a - 3 \right) &= i^{ 0 } \\ \chi^A\left(((2a + 1)b + (2a - 2)\mu_3)c + (a - 2)b + 1 \right) &= i^{ 0 } \\ \chi^A\left((((2a - 1)\mu_3 + (2a - 1))b + (2a - 2))c + ((3a + 2)\mu_3 + (3a + 2))b + 1 \right) &= i^{ 0 } \\ \chi^A\left((((2a - 3)\mu_3 + (a + 1))b^{2} - 3b + (-a + 3)\mu_3 - a)c + ((a + 3)\mu_3 + 3)b^{2} + ((a + 4)\mu_3 + (2a + 1))b + (3a - 3)\mu_3 + 4a \right) &= i^{ 2 } \\ \chi^A\left((4\mu_3b^{2} + ((2a + 4)\mu_3 + (3a + 4))b - a\cdot \mu_3 + 4a)\cdot c + 4\mu_3b^{2} + (4\mu_3 + (2a - 3))b + (-2a - 3)\mu_3 \right) &= i^{ 0 } \\ \chi^A\left((((3a - 2)\mu_3 + 4)b^{2} + (a - 2)b + ((4a + 2)\mu_3 + (2a - 2)))c + ((a + 1)\mu_3 + (2a - 1))b^{2} + ((3a + 3)\mu_3 + 2)b + 4a\cdot \mu_3 + a - 3 \right) &= i^{ 0 } \\ \chi^A\left((((a + 4)\mu_3 + (a + 4))b^{2} + (2a + 4)b + (-2a - 2)\mu_3)c + (-2\mu_3 - 2)b^{2} + (2a - 1)b + (4a + 3)\mu_3 \right) &= i^{ 0 } \\ \chi^A\left(((2a\cdot \mu_3 + 2a)\cdot b^{2} + (2a + 4)b + (4\mu_3 + (4a + 4)))c + ((2a - 2)\mu_3 + 4)b^{2} + ((2a + 1)\mu_3 + 2a)\cdot b + (2a - 2)\mu_3 - 2a + 3 \right) &= i^{ 2 } \\ \chi^A\left(((4\mu_3 + (2a + 4))b^{2} + ((a + 2)\mu_3 + (3a + 4))b + ((-3a + 4)\mu_3 + 2a))\cdot c + (2a\cdot \mu_3 + 4)b^{2} + (2a + 1)b + \mu_3 + 4 \right) &= i^{ 0 } \\ \chi^A\left(((-\mu_3 - 1)b^{2} + (2a + 2)b + 4\mu_3)c + ((2a + 4)\mu_3 + (2a + 4))b^{2} + 2a\cdot b + 1 \right) &= i^{ 1 } \\ \chi^A\left((((3a + 4)\mu_3 + 3a)\cdot b^{2} + ((2a + 4)\mu_3 + (3a + 4))b + (2a\cdot \mu_3 + 4a))\cdot c + (4\mu_3 + 4)b^{2} + (4\mu_3 + (2a + 3))b + (-2a - 1)\mu_3 + 4a \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 } \end{array} \)

Inertia Polynomial

The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + -116734822955810339022788239892a - 191408607873699715846936539480 x^{47} + -151174821443575683392591204412a - 150330813421666723695115484160 x^{46} + 7080755647842594109106641144a - 450617874900493578507713181248 x^{45} + (586394198591908393808879859592a + 72164040971130333449325882168 )x^{44} + -303167068249133423340321960672a - 412119990929078857190066563172 x^{43} + (-423695499410085481801120102128a + 167602899380489587137339657168 )x^{42} + -458604578162645012825590518344a - 503793532906816824514147978424 x^{41} + (-10901565668603586394020941480a + 285447552803750867169911837152 )x^{40} + 347255963416140145363746287840a - 431066051487085599265007879168 x^{39} + -611369848731241765070351869752a - 374120362895478614100747792300 x^{38} + (-195761168489465419627932540300a + 229795655650131037976539431456 )x^{37} + (-73981771099425197271894764096a + 462970695203766309329128764488 )x^{36} + -464554451403011203839678535304a - 287046709572725311435491067656 x^{35} + -478507116617967230960101683340a - 110192197184480290737502522752 x^{34} + -216268085387456875716423939160a - 305559675068802053445147489272 x^{33} + (56063598405103511594915724332a + 184260324509254803942256025416 )x^{32} + (75340153463379895215974830928a + 302055912631919658011745987488 )x^{31} + 58771333166096581044584268492a - 309613724298726213291506093036 x^{30} + 447091931946467099077601231504a - 584973209164174310591795276856 x^{29} + (460280562771368228824626326696a + 94970427647338024295348400144 )x^{28} + -421568490618686845584035459344a - 473241948687040958239711227152 x^{27} + (-106101168426576763939970520472a + 553551929070551611058002556208 )x^{26} + (-487617911271718342913104389456a + 195131817219953514836155291000 )x^{25} + (-562615804814241504050859391530a + 313105909419575980420412570504 )x^{24} + (-10925523380624822666527800800a + 341772946836192118697839110888 )x^{23} + (-551696540898785291756116775480a + 60478866902130366124232691832 )x^{22} + (568823890298653078706649191968a + 583792131605516845788637589328 )x^{21} + -71916873509880476609284363208a - 272743053928983902058097832856 x^{20} + (520961850600971134559430053052a + 413724045958601473147197559936 )x^{19} + (580313421926159257120012390528a + 48284006234299325555789602976 )x^{18} + -210613234662268451935702719848a - 204994385469198914912535627696 x^{17} + (-462544988857276508556885624248a + 57997380945279895828990693552 )x^{16} + (348776007907314487947896808160a + 318406070009931215633576547024 )x^{15} + -79489354686272976431390053468a - 333177937916849832266673702272 x^{14} + -146060746958507057664034816968a - 74926148943693260086515139672 x^{13} + (360534984307433071769618480704a + 390511767600820309794304681244 )x^{12} + (-311078330328262817674774435144a + 415982308613344913863543829600 )x^{11} + (-242548356704347316822137085656a + 493678354532074189611009134408 )x^{10} + (315952944993230177277537442376a + 508436539186543430600278631088 )x^{9} + (-571727530536987387066194962416a + 594541777616567182551955692376 )x^{8} + (-177318031011542263042193946528a + 558328774177103060218194986560 )x^{7} + (384943874323948465882567323132a + 566759606067524056369089434504 )x^{6} + (379403968899327883451423631480a + 522888949086009103244188591840 )x^{5} + (-523226541949375692857414523664a + 506023041426534673913214273728 )x^{4} + (586549891632168850954456008320a + 391740866967400532585611310208 )x^{3} + (-25346739273682899594868588336a + 355401238020195403480187062960 )x^{2} + 568854334221025378099086199920a - 579601551631774983747819960080 x - 629807102457210958479944757082a + 198960360512984825844442444666 \)
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