ex.24.7.1.55_79_103.c
Base Field
\(F = \mathbb{Q}_2\)
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
\( v(N) = 7\)
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 \(x^{3} + 2 \)
Inducing Field
The inertial type \(\tau\) becomes reducible over
\(K = L(c)\), \(c\) a root of \(x^{2} - b^{3} x + (3b^{2} + (3\mu_3 + 3))b \)
Underlying Character
Character \(\chi^A:\mathcal O_K^\times \to \mathbb C^\times\) with the following properties:
Order
4
Conductor exponent
11
Values on generators of
\((\mathcal{O}_K/\mathfrak p^{ 11 })^\times/U_{\mathfrak{p}^{ 11 } }\)
:
\(\begin{array}{l}
\chi^A\left(\mu_3c + 1 \right) &= i^{ 2 }
\\
\chi^A\left((-\mu_3 + 3)c + -\mu_3 + 3 \right) &= i^{ 0 }
\\
\chi^A\left((2\mu_3 + 2)b^{2}c + 1 \right) &= i^{ 0 }
\\
\chi^A\left(((2\mu_3 + 2)b^{2} + 2\mu_3)c + 4\mu_3 + 1 \right) &= i^{ 2 }
\\
\chi^A\left(((3\mu_3 + 3)b^{2} + 4\mu_3)c + 1 \right) &= i^{ 1 }
\\
\chi^A\left(((\mu_3 + 1)b + 2)c + (2\mu_3 + 2)b - 3 \right) &= i^{ 0 }
\\
\chi^A\left((3b + 2\mu_3)c + 2b + 4\mu_3 + 1 \right) &= i^{ 0 }
\\
\chi^A\left((\mu_3b^{2} + 4)c + 1 \right) &= i^{ 0 }
\end{array}
\)
Inertia Polynomial
The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + 425541065674964083844296654248x^{47} - 293922586372195556873163317108x^{46} + 208020178044503327051062937672x^{45} - 103162378658804955690726618484x^{44} + 321371221333412531268670694192x^{43} - 313410746724552304098866941832x^{42} - 6813454099040849630771172416x^{41} + 38143533461783126205332474160x^{40} - 358616770170067520439694915808x^{39} - 89314989171033227132064304584x^{38} + 472039964740469660294283936112x^{37} + 121234994773993020306134164776x^{36} + 224985170968364489625522493344x^{35} - 430495259104046564229926536216x^{34} + 316195405143687051241901142128x^{33} - 601929706974295527751471899184x^{32} + 447467695321440952425416681616x^{31} + 4785485670402192179580027584x^{30} + 339163934331649389469468152296x^{29} + 126688375333198426648151458512x^{28} - 394315558889640056282015799104x^{27} + 477285724608447582956688851184x^{26} - 260938690996481043413200623216x^{25} - 78597545576050950588863485668x^{24} - 562730399958303912169176283632x^{23} + 71876843418068660347028253848x^{22} - 158323810551699369113694742544x^{21} + 552579489441453624404136940888x^{20} + 556761085319125807932887601280x^{19} + 297717152188417994013218879408x^{18} + 19147887005665822865933556128x^{17} + 564512767272096080839963226080x^{16} + 531844706457572537127714110976x^{15} - 418978224737480029766146671568x^{14} + 113575000418993817716767420640x^{13} + 378203208077916583374002808448x^{12} - 410025451386070575565501446560x^{11} - 448223779135086529709464392432x^{10} - 163570191739654460000515893024x^{9} - 256485750106060155495179347168x^{8} + 505632308725025628946435826976x^{7} + 481744402246587381388394793696x^{6} - 99618692227750374389356102064x^{5} + 183777504416380365088610775104x^{4} - 338230468708181370908159037632x^{3} + 588460180730807352954360140608x^{2} + 211104833363790696424665377248x - 387978783409005373455035924236 \)