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ex.24.10.1.31_63_95.a

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 \(x^{2} - 211275100038038233582783867563b^{3} x + (-211275100038038233582783867563\mu_3 - 211275100038038233582783867563)b \)

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

Inertia Polynomial

The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + -23290113418896581130284151300a - 379227890249476545421003768528 x^{47} + (245050602153198349197745451196a + 466125648713318656243749664256 )x^{46} + (302222498098851435454305245608a + 510776759853027452311281905944 )x^{45} + (-326200324330109348835355471572a + 407549671676325237194567436432 )x^{44} + -40938181563727361320929289552a - 391506527862167149466893787876 x^{43} + -496001743788742300773131812484a - 101509380637111629118928560056 x^{42} + -501676697382376500658797367080a - 605822467305785055049907390216 x^{41} + (11074246289385870210832592264a + 631785617983300764332570557192 )x^{40} + (146424603838143975792849661520a + 249642829273094729012913538032 )x^{39} + (535859708345925981811728671048a + 397687040006216883483203708324 )x^{38} + (-161125454291678884443189530776a + 494183306642993439749944802824 )x^{37} + (84993549335862606017189527996a + 287147218685836842014523277600 )x^{36} + (511406438651123503147839820928a + 120793753376432859150199770760 )x^{35} + -185208444334522203808713304972a - 212366843884486662736762576992 x^{34} + (230046406592215053205352440616a + 362944112322130031071450661128 )x^{33} + (-214332935174032023742424849532a + 281323245947935341949681367160 )x^{32} + (4916638794781307936706133540a + 387319309416431428331896025136 )x^{31} + (142359753511229523324904340936a + 1179268699634939375912994244 )x^{30} + (-133327623911812795233496474984a + 330075035952996449517397781488 )x^{29} + (315471133193625847107361182552a + 541080212577004524215166079088 )x^{28} + -340946770617454314720260730032a - 623309500664419283805809999296 x^{27} + (235799419498564613841218832752a + 456920424371549862794017565480 )x^{26} + -389614025403904964832409192192a - 296905652324438863820676914880 x^{25} + (-311652539040374487033177480010a + 65793893021393836636526906464 )x^{24} + (77828650117608550178276646544a + 74337600117265277404104199800 )x^{23} + (214323166213291984794999888336a + 526408190842105553266809465704 )x^{22} + (-239335733666493338365117073416a + 336970392874253875802626920944 )x^{21} + -574894925629423108624755388888a - 471722326741161871805822036672 x^{20} + (424255330657508477856486237428a + 126385045272087085697510399696 )x^{19} + 164238598613280829198726668416a - 493832104674473214182295612520 x^{18} + -294967540245351632923259363208a - 291356796675808913900515955264 x^{17} + -405859710698886838059226700720a - 562342587097389657867588948224 x^{16} + -25845209438209466569281316656a - 240615154130905218717105009680 x^{15} + -31907236975082118680377370028a - 582759880333156490612544091952 x^{14} + (341762550900465050632367350432a + 68460374524800328375696519248 )x^{13} + -545173956682657295362718211840a - 63638441565522118767862697620 x^{12} + 473095086562831218579494158424a - 195817073782313300784212512992 x^{11} + 153842880272531488466145288848a - 238163632612802790769831257944 x^{10} + 615770955134941026916110936a - 524951273221978243079632133072 x^{9} + 411795303513635390894660872104a - 393437757991107162468047884856 x^{8} + -135166261880244919817182395280a - 606315045565186575978969080744 x^{7} + (-492449812490090617470006907244a + 166825967714660301882008761232 )x^{6} + (359613914886524195369593440a + 543844789136939640580913686832 )x^{5} + -305348673437867294454692091600a - 261843158288674426854389448608 x^{4} + (388991500364190400666716133920a + 493386345453738465975254061664 )x^{3} + (210225916464470691895243182360a + 498509089686820972826464535904 )x^{2} + 136025928204983754292272328600a - 625123872920443552444703989536 x + 464454103950716701678285661896a + 381948062081823493024745466282 \)
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