← Back to 2.1.2.2a1.1

ex.24.10.1.127_255_383.c

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

Inertia Polynomial

The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + (-292175201196363914731666799768a + 384591559050319454541746362424 )x^{47} + 199345107476208458067660610960a - 438207211123702788432710001860 x^{46} + (-372561157702697453617160316124a + 507751944994001400867842879952 )x^{45} + 192488908054421666340981910068a - 510483902802020023535637242552 x^{44} + (356182974563304633357704962272a + 282901752151032880548226061440 )x^{43} + 555865152842117596334066781096a - 387995594121016336923013289124 x^{42} + (437068687317380748479287088128a + 561584937362386502291211056024 )x^{41} + (163048864955106290181582486984a + 150267734309620718009393807908 )x^{40} + 297167200519466735866856643992a - 494386816493848191242439022240 x^{39} + (-513789610912454534724151453448a + 629745856243354241602866873936 )x^{38} + 595630350622266521264416426604a - 143364740779893170740505423296 x^{37} + 525068850648466862591042436940a - 551983932316195885861481617628 x^{36} + (523313531665286513226075569480a + 109851078712780183595649925456 )x^{35} + -268422339223067427523296484304a - 390290224659999749129557534384 x^{34} + -620435063693520676720010882496a - 141078259004156056551719592824 x^{33} + (113189690179826821051450740092a + 138448901292060865472244864528 )x^{32} + (-602165005652724586628285274976a + 406769808787592787556622919328 )x^{31} + (244403870212999534216254348400a + 262508635455909754190884362848 )x^{30} + (542736818595966065466146608792a + 381536379383959148788786799088 )x^{29} + -97091237106126568255585558816a - 86933468037242677718588812904 x^{28} + (-325110334083144491424131038000a + 546658754420052944252748663424 )x^{27} + -154425086526365472992092948920a - 548695621620617413523181558728 x^{26} + -439290264839240149658321842120a - 12097251892088783309980763112 x^{25} + (-5013524324312560423047845378a + 221952871747823086750690468976 )x^{24} + (49729952241218180185119057080a + 107713269045276805152963316176 )x^{23} + 336661838603834560441205803908a - 235780977675100474284558087280 x^{22} + (297671433532480357689060931224a + 461730426456160955592022282504 )x^{21} + (491168479384695257012094210384a + 508446176628732861016440738424 )x^{20} + (72937825514755578044032128704a + 434307317322581174869937569232 )x^{19} + -206300663113320207901486936516a - 471743288416104035226347157728 x^{18} + 599940358774685697964770541592a - 573904408394503207816434291168 x^{17} + (-179197470955969302526985111044a + 522890244102221480478591664704 )x^{16} + (-345510701098770218241534713488a + 541067685429106028125200229984 )x^{15} + (615516206706689566724830320752a + 369017729499016186667202908464 )x^{14} + -442150133889251381130563410296a - 180597343170786667212584343176 x^{13} + -371611473987473629581208290760a - 130070903846742846563142811248 x^{12} + -434012788943062369499246610032a - 127879619259862385462987123696 x^{11} + (64818069511652236652681785784a + 559610810005956974066225372384 )x^{10} + (-383725841601644521468973831880a + 589925408596442087315518011680 )x^{9} + (436661870650785293068288147432a + 493646790944510714451235746888 )x^{8} + 402615202542193452601627736336a - 575748048092004521541337702672 x^{7} + (439463454160848530945433667296a + 558582289954287129547624308768 )x^{6} + 491332332236189182838834145488a - 566797287532103645673470349424 x^{5} + 631247101136665415618276429032a - 567696329905818308792369360896 x^{4} + 413095675616716104873250507888a - 325075108986675894845342196992 x^{3} + (553997195821051876182740991320a + 532539043670315547870971080576 )x^{2} + 536254067031550556200459296792a - 296725127709680738435612022576 x + 502087502348184878713763391198a + 261503391421578735504968382742 \)
← Back to 2.1.2.2a1.1 Summary