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ex.24.10.1.131_259_387.a

Base Field
\(F = \) 2.1.2.3a1.1 \( = \mathbb{Q}_{ 2 }(a) = \mathbb{Q}_{ 2 }[x] / (x^{2} + 2 )\) 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 \(x^{2} - b^{4} x + 3b \)

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

Inertia Polynomial

The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + (179899114500228288783504819408a + 380101234048457370666818123832 )x^{47} + -246878885783893065442295919380a - 434283294100576347564459671184 x^{46} + (396642219208095480438270672116a + 155799774075126965237679021104 )x^{45} + 315951002637267459891477665352a - 312992642490590805289963731508 x^{44} + (-78695239192340864716535393960a + 580604928358042515849520193856 )x^{43} + 548368372371829190225883576004a - 332362879494340836901545876836 x^{42} + -362108920907466515433738972528a - 87347770021616728431041978240 x^{41} + (151038600252684303506596974172a + 583179828134659717987546246020 )x^{40} + -381145537486319578239752109064a - 557123436471951342561149321648 x^{39} + -493759244307312149434288344196a - 297571905737983671276628304024 x^{38} + (-22558154064814776662813640092a + 399003750829283596262754446248 )x^{37} + (572065011683534115401425401652a + 92664934475252653404868313164 )x^{36} + (-385512566921538503007193345296a + 429139105342535431164497787072 )x^{35} + (431146831581798056239018962228a + 108184189264265690924303393680 )x^{34} + (459771726692870745185431099048a + 372742248817800137178102870960 )x^{33} + (270952868205775568098517568960a + 367433075625365769380427568904 )x^{32} + -186219827983439235568101205248a - 459442125101200124542647076096 x^{31} + (-469432139844272133727739387376a + 525017161802060732315392295248 )x^{30} + -242355937773812085109800423968a - 535434838460942504388945578072 x^{29} + (-156506562673253709327773843392a + 111255833250954899864375019160 )x^{28} + (399333958351985714211341978976a + 396332664534095158433707372912 )x^{27} + (-517483715499855807906048936260a + 68127082542903259803633260512 )x^{26} + -532819038559666439843746428000a - 363998577222866005800706185112 x^{25} + (463235014411546101125421972522a + 582928943133981779494957066708 )x^{24} + -591224001747745208918078902040a - 188607847427063302482831697744 x^{23} + 205429152642988944048459711872a - 197147950381053129757516247608 x^{22} + 631377531530770270885650428376a - 66078584041042855655710202536 x^{21} + (-331224660484457443964265296036a + 282980072596624692086950086632 )x^{20} + (3732114723496225880975761760a + 339830407788192821901974301424 )x^{19} + (-426116342665688827329223757316a + 533096300072019867039426776208 )x^{18} + (51398467837644391192900822272a + 531260105994353959638917545456 )x^{17} + (-582170207135972101264131694828a + 596619771879324567631977379824 )x^{16} + -202760071506218517659928747728a - 564448392541894474499364622272 x^{15} + -228297240711048110797993386872a - 372871887925647496420870178744 x^{14} + (410790731304879461689303455920a + 154672271044181855690636697992 )x^{13} + (403002050314049241570066832368a + 527202674736801015140126463136 )x^{12} + -306396346141750893581929303824a - 334565628456028163764717555360 x^{11} + (-80183956223879958423487030160a + 157136976666896661804224012648 )x^{10} + -213598696522208565521213266304a - 242168339655550821036961150416 x^{9} + 222480121610128219199152861896a - 531178303564947588248564564896 x^{8} + (-542869673255105546684135392a + 249240066107454071591690325648 )x^{7} + 264355025416083227338714741056a - 34244379774538878394686541824 x^{6} + (-455608271071659324828596841688a + 441255601334714737394268340144 )x^{5} + 472830069043352871466862433624a - 429632564909577808278768218928 x^{4} + (189775882841034049657455530416a + 297214109686559699087855298688 )x^{3} + (-329414084910087294470965811944a + 242376184362463347505107072872 )x^{2} + 166760648851795210390261223032a - 568785669886258680328887877616 x - 557970864129412083038401803236a + 613580311438604250328553350426 \)
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