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ex.24.8.1.31_63_95.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
\( v(N) = 8\)
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 \(x^{2} + ((a + 1)b^{3} )x + (-\mu_3 - 1)b \)

Underlying Character

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

Order
4
Conductor exponent
14
Values on generators of \((\mathcal{O}_K/\mathfrak p^{ 14 })^\times/U_{\mathfrak{p}^{ 14 } }\) :
\(\begin{array}{l} \chi^A\left(\mu_3c + 1 \right) &= i^{ 1 } \\ \chi^A\left(((2a - 2)b^{2} + (-2\mu_3 - 2)b + ((4a + 2)\mu_3 + 4))c + ((a + 1)\mu_3 + (2a + 3))b^{2} + ((a + 1)\mu_3 + (2a - 2))b - 2a\cdot \mu_3 + a + 1 \right) &= i^{ 2 } \\ \chi^A\left((((a - 2)\mu_3 + 3a)\cdot b^{2} + (4\mu_3 + 2a)\cdot b + 4a\cdot \mu_3)c + ((2a - 2)\mu_3 + (2a + 2))b^{2} + 4b - a\cdot \mu_3 - a + 1 \right) &= i^{ 0 } \\ \chi^A\left(((\mu_3 + 1)b + (2a + 2))c + ((3a - 2)\mu_3 + (3a - 2))b + 1 \right) &= i^{ 0 } \\ \chi^A\left((\mu_3b^{2} + ((2a - 2)\mu_3 + (2a - 2))b + 4)c + (2a + 4)\mu_3b^{2} + (2a\cdot \mu_3 + 2a)\cdot b + 1 \right) &= i^{ 0 } \\ \chi^A\left((-b + (2a + 2)\mu_3)c + (a + 2)b + 1 \right) &= i^{ 2 } \\ \chi^A\left((((2a + 4)\mu_3 + 2a)\cdot b^{2} + 2a\cdot b + (4\mu_3 + (4a + 4)))c + (2a + 2)\mu_3b^{2} + (3\mu_3 + (2a + 4))b + (-2a + 2)\mu_3 - 2a - 1 \right) &= i^{ 2 } \\ \chi^A\left((((a - 2)\mu_3 + (2a + 2))b^{2} + ((3a + 4)\mu_3 + 3a)\cdot b + (3a + 2))c + ((a + 3)\mu_3 + (3a - 3))b^{2} + ((3a + 3)\mu_3 - 2)b + (4a + 2)\mu_3 + a + 3 \right) &= i^{ 2 } \\ \chi^A\left(((3a\cdot \mu_3 + 3a)\cdot b^{2} + (2a + 4)b + (-2a + 2)\mu_3)c + (2\mu_3 + 2)b^{2} - 3b - \mu_3 \right) &= i^{ 0 } \\ \chi^A\left((((2a + 4)\mu_3 + 2a)\cdot b^{2} + ((2a + 4)\mu_3 + (3a + 4))b + (3a + 4)\mu_3)c + 4b^{2} + (4\mu_3 - 1)b + (2a + 1)\mu_3 \right) &= i^{ 0 } \\ \chi^A\left(((3a\cdot \mu_3 + (a + 2))b^{2} + ((a - 2)\mu_3 + 2a)\cdot b + ((a + 2)\mu_3 + (3a + 2)))c + ((2a + 2)\mu_3 + (a - 3))b^{2} + ((a + 1)\mu_3 + 2a)\cdot b + (a - 3)\mu_3 - 3a - 3 \right) &= i^{ 2 } \\ \chi^A\left((-\mu_3 + 3)c + -\mu_3 + 3 \right) &= i^{ 0 } \\ \chi^A\left(((a\cdot \mu_3 + (a + 4))b^{2} + ((2a + 4)\mu_3 + (3a + 4))b + 2a\cdot \mu_3)c + (4\mu_3 + 4)b^{2} + (4\mu_3 + 1)b + (2a + 3)\mu_3 + 4a \right) &= i^{ 0 } \\ \chi^A\left(((-\mu_3 + 3)b^{2} + (3a\cdot \mu_3 + (a - 2))b + (4a\cdot \mu_3 - a + 4))c + (-3\mu_3 - 3)b^{2} + ((a - 2)\mu_3 + (2a - 2))b + (2a - 2)\mu_3 + 4a + 1 \right) &= i^{ 3 } \end{array} \)

Inertia Polynomial

The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + (-265082562469940717161261658044a + 272816921959766874900991036952 )x^{47} + -579947053705962465352214022536a - 285755211213729860949801194368 x^{46} + (-607687461345492863643981847660a + 55263271982534510453069345040 )x^{45} + (-2307547788607404013595938764a + 466163086896803000097541125016 )x^{44} + (347932269718636218038556470436a + 605473641466121756536091014520 )x^{43} + -187958864496227899528173648200a - 70986907234264842652609122340 x^{42} + (118047560715341283645234634520a + 84637284631299538388139175860 )x^{41} + (507146269717376047335110844384a + 559040458766152674931575864972 )x^{40} + (45099524970357578894340254600a + 570737024834212628384234038336 )x^{39} + (-408243430254557688853714806104a + 510866859458877810969433813228 )x^{38} + -166588245433592004079137611896a - 5674886848193225041718697268 x^{37} + 198205648757354023865401369224a - 604149920841756513559822882508 x^{36} + (400888820349929701471365251856a + 486802674771058141391262788904 )x^{35} + (434634085961788758912043565824a + 349492478083033180351250404448 )x^{34} + 29889035031741756537679112384a - 394715912458538007676698231232 x^{33} + 460813310394375256430355923420a - 153931858931259140263754195316 x^{32} + -454877113028405546662454656288a - 205057320597588227323080597772 x^{31} + (-350975573291962908104803149116a + 135586862670053569296058983764 )x^{30} + (-548208023990464584924124971604a + 302759748521538873500376834632 )x^{29} + 10588938318347641487578673540a - 96408970520699119268197685552 x^{28} + (123545112331737061495453859520a + 52834768954951637200115234512 )x^{27} + (404869961289742208381544768584a + 334864199543932445075388199896 )x^{26} + (111921488577532789244538163340a + 545647204414739768776987366320 )x^{25} + 274138238336797497401086720530a - 356951336489520790523142080888 x^{24} + (-246085313089392215309767776896a + 224214389268649206571498390592 )x^{23} + (-169941992700004793816397195728a + 56601740008989672500326769768 )x^{22} + 505187361094259728862298447656a - 83960182898953839818670317880 x^{21} + 177037347948335213501579899400a - 554785393424392907727470975336 x^{20} + (-319861794931466831825992909624a + 188177641399628894252132594344 )x^{19} + (168301060077772525005202035500a + 52871809739939264570888288728 )x^{18} + (-240132823891529405411050615500a + 609058033155707177873169144136 )x^{17} + (283372818199386111531613267516a + 7592135702151798046318653000 )x^{16} + (-92418783410031986272847053216a + 419126138371654664628964199792 )x^{15} + 297406454210920891921534884644a - 405870491509666883059772860000 x^{14} + -87331435850566190006442234412a - 237640999631421913619574603856 x^{13} + (-14534755441635191334227274196a + 257132379671867273874805643236 )x^{12} + (-427912109794480056774908782712a + 419145176360822880982898400928 )x^{11} + 379069378721558581059476639120a - 330860669685414341790642669760 x^{10} + 93913594569607947368317563504a - 457387700892365129709236494160 x^{9} + (-451517772517366445037037415644a + 110252167467691073554461809360 )x^{8} + (469082897037859681874224980820a + 232321694308494888160186463976 )x^{7} + (-284128295728076477226793070084a + 353153692653396983547601688168 )x^{6} + (-526513300906136971613951533408a + 42128818842819722105328816024 )x^{5} + 547345340835000374984450001592a - 224381335312285596211799896824 x^{4} + 105943181917035299050918441648a - 353149680138143681315641600944 x^{3} + (182166547361635635564292533312a + 176646397353404023964761823040 )x^{2} + (-575831567557898025402918393784a + 97532070596253314071853094040 )x - 570126353496244081418935338222a + 491269493140565956305567819162 \)
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