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ex.24.10.1.127_255_383.b

Base Field
\(F = \) 2.1.2.3a1.2 \( = \mathbb{Q}_{ 2 }(a) = \mathbb{Q}_{ 2 }[x] / (x^{2} + 5\cdot 2 )\) 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 \(x^{2} + 253530120045645880299340641075b^{4} x + b \)

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

Inertia Polynomial

The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + 193493051884295649913944560368a - 166048338844745378223307361960 x^{47} + (409839795844786379381377429284a + 464021029977424960184141884256 )x^{46} + (617401183343046850652192932844a + 78449896606820950422205128656 )x^{45} + (-426477862530820437825722743400a + 308040679870718434220051712004 )x^{44} + -494271829835042927508903926368a - 505018028137837103443002104336 x^{43} + -396675756816040383793158498676a - 434563425957192621917608352516 x^{42} + (-108055210576792506285272646248a + 279042444889335383058318519536 )x^{41} + 119375011738536020484048178560a - 105805426720354645715501939428 x^{40} + (329333284428451510335477077432a + 357539764478740051864968061664 )x^{39} + (-367361984395953701258074794164a + 22047558935058280872899956352 )x^{38} + (200482799523974535354038326044a + 247684816749750007714528457808 )x^{37} + (11079833995066844631470724356a + 497188016652788055904807627668 )x^{36} + -389467001399986553609273599528a - 54124534213052335599599138112 x^{35} + (341180788602615713239118910420a + 390833620747112258779902033080 )x^{34} + 222938352763423018325320866744a - 279396989485513639512598345872 x^{33} + (548010899658150183970415846436a + 183952575085199995512345281096 )x^{32} + -587155075952831680689983593504a - 218120642626221143568964542112 x^{31} + 537897182913994463764135771520a - 235011917486269876415118857376 x^{30} + (-58961918714390130090726472648a + 495422635741114286798319310904 )x^{29} + -241878450204286007343552653460a - 122016168515948807634112892832 x^{28} + 159998714180975275852286525872a - 246476159353787980601042007088 x^{27} + (-441933095969051781456147122892a + 558253179110804174898712517120 )x^{26} + (34530173552785098800952836360a + 273508989428479827646059845888 )x^{25} + (368721088160145720893226372766a + 532917909200206962156155999280 )x^{24} + (18528248928428132180143296168a + 259839876222157337331810608832 )x^{23} + -617396671404631860561176630688a - 230870607830425381763341907016 x^{22} + 195013218970053773104053988384a - 352022270510415309267995661864 x^{21} + (-483266978896575066691085506724a + 264239479110885815550453124752 )x^{20} + (-414416426203150412208475871216a + 556222053848692588124768156624 )x^{19} + (-408394754002129265855910792276a + 283381761151067568618417899928 )x^{18} + -466226218375294978828475415568a - 369750089450101151936524710400 x^{17} + (-80409460356367435364846192100a + 632280805390188546687815551680 )x^{16} + 188485607824933369980905350240a - 542389678785457019326079489856 x^{15} + -556341782837647717475424742472a - 182263150510448535799092608120 x^{14} + (18445627787177818828832095552a + 138135677308757378244989288120 )x^{13} + (-2564225776595702117078387816a + 361603313104803979028832711712 )x^{12} + (-413387721645658852351357551696a + 326039014716473328311607669712 )x^{11} + (335807201607027856368352203344a + 548814189502157583359856213944 )x^{10} + 497867984622114368197609875184a - 84263424227140976851593195888 x^{9} + (319534178422501372324071423896a + 599207604099166572840336884920 )x^{8} + -232976719768018939888047738080a - 547897511493470968938156153904 x^{7} + -66835912352186554069678545424a - 2047549567121597825761430688 x^{6} + -129590620753038354702985734664a - 62929716338728328452703284880 x^{5} + (322457980439153822150539592752a + 97432153778943128773511223704 )x^{4} + 572259590138138012114768218800a - 389996955916698767691863654272 x^{3} + (-131618264789765597833816375280a + 506334925806968005121362695320 )x^{2} + (52216886571827839244292757248a + 375762939381508962836256735344 )x - 576766171741571584515236507328a - 624722455376975081451093794466 \)
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