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ex.24.10.1.33_67_101.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 \(x^{2} + ((70425033346012744527594622521a - 70425033346012744527594622521)b^{3} )x + ((-105637550019019116791391933781a - 211275100038038233582783867562)b + (-140850066692025489055189245042a + 140850066692025489055189245041))b \)

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^{ 0 } \\ \chi^A\left((((2a + 2)\mu_3 - 2)b^{2} + (4\mu_3 - 2)b - 2\mu_3 + 4a + 4)c + ((3a + 3)\mu_3 + 2a)\cdot b^{2} + ((a - 3)\mu_3 + (a - 3))b + (2a + 4)\mu_3 - a + 1 \right) &= i^{ 2 } \\ \chi^A\left((((a + 4)\mu_3 + (2a + 2))b^{2} + ((2a + 4)\mu_3 + 4)b - 2a\cdot \mu_3)c + (2\mu_3 + 2a)\cdot b^{2} + (2\mu_3 + 2)b + 3a\cdot \mu_3 - a - 3 \right) &= i^{ 2 } \\ \chi^A\left(((3a\cdot \mu_3 + (a + 3))b^{2} + ((2a - 2)\mu_3 + (3a - 1))b + (a\cdot \mu_3 + 2a))\cdot c + (\mu_3 + 3)b^{2} + ((3a + 1)\mu_3 + (2a - 1))b + (-a + 1)\mu_3 - a - 1 \right) &= i^{ 0 } \\ \chi^A\left(((2a + 1)b^{2} + 2b)\cdot c + 2a\cdot b^{2} + 2a\cdot b + 1 \right) &= i^{ 0 } \\ \chi^A\left((((2a + 4)\mu_3 - 3)b^{2} + ((2a + 4)\mu_3 + (a + 4))b + ((4a + 4)\mu_3 - 1))c + ((2a + 4)\mu_3 + (2a - 3))b^{2} + (4\mu_3 + 3a)\cdot b + -2a - 1 \right) &= i^{ 0 } \\ \chi^A\left(((2a + 2)\mu_3b^{2} + ((3a + 4)\mu_3 + (2a + 4))b + ((3a + 4)\mu_3 + 4a))\cdot c + ((a + 2)\mu_3 + 4)b^{2} + ((2a + 3)\mu_3 + 4)b + (-2a - 3)\mu_3 \right) &= i^{ 0 } \\ \chi^A\left((3a\cdot \mu_3b^{2} + (a + 4)\mu_3b + 2a\cdot \mu_3)c + (a - 2)\mu_3b^{2} + (2a + 1)\mu_3b + (-2a - 1)\mu_3 \right) &= i^{ 0 } \\ \chi^A\left(((-2\mu_3 + a)b^{2} + (4\mu_3 + (3a + 4))b + ((-a + 2)\mu_3 - 2a + 4))c + ((a + 2)\mu_3 - 2)b^{2} + ((2a - 1)\mu_3 + 4)b + \mu_3 + 4 \right) &= i^{ 0 } \\ \chi^A\left(((4\mu_3 + 4)b^{2} + (2a + 4)b + (4\mu_3 + (4a + 4)))c + ((a + 2)\mu_3 + (a + 4))b^{2} + (\mu_3 + (2a - 3))b + (-2a - 2)\mu_3 - 2a - 1 \right) &= i^{ 2 } \\ \chi^A\left((((2a - 2)\mu_3 + (2a + 2))b^{2} + (3a\cdot \mu_3 + (a + 4))b + (3a\cdot \mu_3 - a + 4))c + (-\mu_3 + 3)b^{2} + ((a - 3)\mu_3 + (a + 1))b + (-a - 3)\mu_3 + 3a - 3 \right) &= i^{ 0 } \\ \chi^A\left(((a + 2)b^{2} + (2a - 1)b + (2a - 2))c + (2a - 2)b^{2} + (3a + 2)b + 1 \right) &= i^{ 0 } \\ \chi^A\left((((a + 2)\mu_3 - 2)b^{2} + ((2a + 4)\mu_3 + (3a + 4))b + (2\mu_3 + (3a - 2)))c + ((2a - 1)\mu_3 + (a - 2))b^{2} + ((3a - 1)\mu_3 + (a + 3))b + (-2a + 2)\mu_3 - a + 3 \right) &= i^{ 0 } \\ \chi^A\left(((2a + 1)\mu_3b^{2} + 2\mu_3b)\cdot c + 2a\cdot \mu_3b^{2} + 2a\cdot \mu_3b + 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} + (156875814450899096738459668296a + 455673895382262948470916784544 )x^{47} + 30874171359416147561574858132a - 569017997773787389528155653972 x^{46} + (-38705102349996970234767670188a + 95732142899583204357292047832 )x^{45} + 140066955587852207990750816320a - 460628187109503189440365570556 x^{44} + 195261119659009738145739887836a - 58477956471211545138257068668 x^{43} + (-413775884914727688306090859536a + 367511673497190845194667139272 )x^{42} + (-284443677563673736391609808432a + 161641843036167493164090181768 )x^{41} + (-323822214870182084469741090484a + 108370548771484728185399011400 )x^{40} + (252909557494215318869674028584a + 578793452533780501004490419376 )x^{39} + 351005062622800243579380742428a - 367027156659675581116356965564 x^{38} + (-306970974111601180021876124348a + 221430560731584645783644396628 )x^{37} + (75110919737814781329033724364a + 327769101093459198408206170976 )x^{36} + (471073987866723265961486282968a + 348919034600719693148699237096 )x^{35} + (176679440160866643723794855276a + 199665780198868189096038186504 )x^{34} + (-376321210672587584429260362668a + 497723890529328902869387059576 )x^{33} + (-602505740374311339580510290508a + 225794532877795749244987084112 )x^{32} + (-11837387728941309966279722100a + 134378888880968981228555101584 )x^{31} + (-359332812896870095466609399648a + 44314550422889695151308691636 )x^{30} + (-367122483612857130139943502156a + 68748456494895325077415155648 )x^{29} + 475546864413578256933862343856a - 496072434920780033029316868520 x^{28} + (563092713971568339192817766936a + 248536454467314531254327302920 )x^{27} + -508017067445850046486134953184a - 430146444592467390692361986164 x^{26} + (341124729209319257793505794124a + 546181173223459501792586720104 )x^{25} + -537002848853929029110281813874a - 623803773607129091587215151876 x^{24} + (-20177597448112525871241155632a + 124217979702272781988808785120 )x^{23} + -349230028860501619901354753444a - 82634227111897296689181504400 x^{22} + -464687149577960312244471236504a - 462643180008852330651788167112 x^{21} + (-483729428111671312929347037436a + 195223384132105571811139874448 )x^{20} + 194166664301180600791957353236a - 308799413561847371062830758376 x^{19} + 283628367480087558953144683072a - 612446581306401354291094572224 x^{18} + (-367535263804590777223901991744a + 354344642109367135372328664192 )x^{17} + (374955999743594485969524068192a + 567258793241513375124024780920 )x^{16} + 409173143175791271730616779576a - 258001234468740014106920506704 x^{15} + -97825756401893017640755039348a - 40025021793549754496263229872 x^{14} + -572675460868299190073889241068a - 119692267943170410328777007440 x^{13} + (340429888105073941421864752632a + 596985845227421880346817730396 )x^{12} + 252515335513080345972240125536a - 460339006879405839649132583936 x^{11} + -114956198147848812793376314152a - 262085032363289164547830799448 x^{10} + (53326932212665952767058830680a + 503514181117526724081185476792 )x^{9} + 582478337302044131160277316736a - 515405547680841938475346156112 x^{8} + -112729892276573208304832064264a - 435336961667771256717009526168 x^{7} + (-176458518319118593520774202748a + 370245065925009629458436667624 )x^{6} + (122971694978436728219467732128a + 34473231237219018834664906536 )x^{5} + (615342568140174473845965790720a + 345922227604999337391100270352 )x^{4} + -633621357930351970965193425288a - 230640077688119951838099994128 x^{3} + (513608497686096856505881949436a + 411247062852985825435388391048 )x^{2} + -382715937476865195659196774160a - 373583747779514194319320533496 x + 95452012891850494180037925434a + 336294854368774191735665429646 \)
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