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ex.24.10.1.127_255_383.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 \(x^{2} + ((a + 1)b^{4} )x + 3b \)

Underlying Character

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

Order
4
Conductor exponent
18
Values on generators of \((\mathcal{O}_K/\mathfrak p^{ 18 })^\times/U_{\mathfrak{p}^{ 18 } }\) :
\(\begin{array}{l} \chi^A\left((-\mu_3 + 3)c + -\mu_3 + 3 \right) &= i^{ 0 } \\ \chi^A\left(((3a + 4)\mu_3b^{2} + 4\mu_3b + (2a + 2)\mu_3)c + 4\mu_3b^{2} + (2a + 1)\mu_3b + \mu_3 \right) &= i^{ 0 } \\ \chi^A\left(((-2\mu_3 + 2)b^{2} + (4\mu_3 - 2)b - 2\mu_3)c + (a + 1)\mu_3b^{2} + ((a - 3)\mu_3 + (3a + 1))b + 3a - 1 \right) &= i^{ 0 } \\ \chi^A\left(((2a + 2)b^{2} - 3b)\cdot c + (3a - 2)b^{2} + 1 \right) &= i^{ 0 } \\ \chi^A\left((\mu_3b^{2} + (2a + 4)\mu_3)c + 4\mu_3b + 4\mu_3 + 1 \right) &= i^{ 0 } \\ \chi^A\left(\mu_3c + 1 \right) &= i^{ 0 } \\ \chi^A\left((b^{2} + (2a + 4))c + 4b - 3 \right) &= i^{ 0 } \\ \chi^A\left(3a\cdot c + 2b + 1 \right) &= i^{ 0 } \\ \chi^A\left((((3a + 2)\mu_3 + 4)b^{2} + (-2\mu_3 + 2)b + (4\mu_3 - 2))c + ((a - 3)\mu_3 + 4)b^{2} + ((a - 3)\mu_3 + (a + 1))b + 2a\cdot \mu_3 + 3a - 1 \right) &= i^{ 0 } \\ \chi^A\left((2a\cdot b^{2} + ((2a + 4)\mu_3 + 2a)\cdot b + 4\mu_3)c + 4b^{2} + (-\mu_3 + (2a + 3))b + 4\mu_3 + 1 \right) &= i^{ 2 } \\ \chi^A\left((((3a - 2)\mu_3 + (3a - 2))b^{2} + ((a - 2)\mu_3 + (a - 2))b + (2\mu_3 + 2))c + ((3a + 1)\mu_3 + (3a + 1))b^{2} + ((3a - 1)\mu_3 + (3a - 1))b + (3a + 1)\mu_3 + 3a + 1 \right) &= i^{ 0 } \\ \chi^A\left(((2a + 2)\mu_3b^{2} - 3\mu_3b)\cdot c + (3a - 2)\mu_3b^{2} + 1 \right) &= i^{ 2 } \\ \chi^A\left(((3a + 2)b^{2} + ((3a - 2)\mu_3 + 4)b - 2\mu_3 - 2)c + ((a + 1)\mu_3 + (2a + 4))b^{2} + ((3a + 1)\mu_3 + (3a + 1))b + 2a\cdot \mu_3 + a - 1 \right) &= i^{ 0 } \\ \chi^A\left(((-2\mu_3 + (3a - 2))b^{2} + ((3a + 4)\mu_3 + (3a + 4))b + ((3a - 2)\mu_3 - 2))c + ((a + 1)\mu_3 + 2a)\cdot b^{2} + ((a + 1)\mu_3 + (a - 1))b + 4\mu_3 + a + 3 \right) &= i^{ 1 } \end{array} \)

Inertia Polynomial

The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + 339916984295338558880293426472a - 474712376197493543101902001160 x^{47} + -556331379208536005141369549612a - 600142037822538778308687846772 x^{46} + (133132220045616018341681163012a + 607493359182661815242693957952 )x^{45} + (470320269814750797699975463744a + 524411920331898260645126343856 )x^{44} + -455252351339387443377617609856a - 70001809024547920089341852320 x^{43} + 164573425224792511921142740640a - 430921049958527897332713343252 x^{42} + (-384536526100367000149890005056a + 502654123303888533665136195736 )x^{41} + (-552280967532606452749592794004a + 297461801675958613973657197692 )x^{40} + -212653260094463396323964014376a - 218177969105334448850855886736 x^{39} + 574751667349684979278406339400a - 147790799454466306171482371528 x^{38} + (-544329522714258188078556249140a + 485726060780117230091025209512 )x^{37} + (-209112521922563985457913820968a + 355466728475122932396972933044 )x^{36} + -621452546730281692176368979672a - 424191997652051488321664420688 x^{35} + (-272306832458959209908231655160a + 443726268525966239065858677336 )x^{34} + 183596024216491337625086283000a - 94771535530619416116030458512 x^{33} + (175987110619860533769126439072a + 548575615113348622823223221376 )x^{32} + -517768952669800415384251901376a - 350256564427337976581552574720 x^{31} + (364375891382138745126743695656a + 548018491206999502921928128960 )x^{30} + -472440160762539486406847375432a - 333415472314337289300535227344 x^{29} + (-350112638540633756511416504596a + 358762388785962190140527128864 )x^{28} + 129855764963809310081238526288a - 292193122534396640499890154768 x^{27} + 128297190969615093644133498424a - 378813391786895465673864009208 x^{26} + (-623390152118039516228883071824a + 82413407157057019421644889704 )x^{25} + 53590482083629423719912764902a - 601596499007468580324363830648 x^{24} + 253943357781956856831668725752a - 621327728990529153814512031376 x^{23} + 138426665831911802539180786420a - 395783337780681558432002501528 x^{22} + (-497554882191811549067982241112a + 75543173898150170087713217416 )x^{21} + (381927263569958583814939232656a + 95180460931538348060039252912 )x^{20} + -447568204224261629115251837232a - 38825680285202747358936304816 x^{19} + (-78893615875335977796438983572a + 4267966367597308372179445600 )x^{18} + (350550271418527922680503566216a + 557942111550794608146492641120 )x^{17} + (37732161087172214984411427148a + 221770967763666156678268130984 )x^{16} + -508063122080938197286548771808a - 56050463647050436013611186848 x^{15} + 241095390128427232982035412168a - 393217535469860893721709743312 x^{14} + -421842965193974550480021781520a - 80625275364127436844985144616 x^{13} + (574988490171588210834529833248a + 546662645912266275718705009688 )x^{12} + (-268027178446762902158562686384a + 388272461714994803497330528496 )x^{11} + (-474562573531365969618767263040a + 123094508406031082060641932736 )x^{10} + (341618479716350690073798565840a + 474916584954800033182409512592 )x^{9} + (-48383314901851126344481136656a + 461968860995506167678376222304 )x^{8} + 100228704417462381291822803312a - 141405563476587589335212802320 x^{7} + -140806637099552249177206679824a - 594364021470314957203075938080 x^{6} + -36295476643026892863976277088a - 526665199345437565140192551472 x^{5} + (-507679602732140741596374125096a + 474087414979191716533445295384 )x^{4} + 187213488836702678522362694576a - 376042409890425036099308885376 x^{3} + -243799616229041383069203168840a - 467576787402796139535240386352 x^{2} + (485761326456883010628241905128a + 331763158458366689218461955744 )x - 597603316954828641667120429522a + 393608177149412080579049112182 \)
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