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ex.24.10.1.33_67_101.c

Base Field
\(F = \) 2.1.2.3a1.4 \( = \mathbb{Q}_{ 2 }(a) = \mathbb{Q}_{ 2 }[x] / (x^{2} + 3\cdot 2 )\) 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 \(x^{2} - 211275100038038233582783867563b^{3} x + (-105637550019019116791391933781a\cdot b - 211275100038038233582783867563)b \)

Underlying Character

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

Order
4
Conductor exponent
20
Values on generators of \((\mathcal{O}_K/\mathfrak p^{ 20 })^\times/U_{\mathfrak{p}^{ 20 } }\) :
\(\begin{array}{l} \chi^A\left(\mu_3c + 1 \right) &= i^{ 2 } \\ \chi^A\left(((2a\cdot \mu_3 + 2)b^{2} + 2\mu_3b + 4)c + (2a\cdot \mu_3 + (2a - 2))b^{2} + ((2a + 1)\mu_3 + (2a + 1))b + 4\mu_3 + 3 \right) &= i^{ 0 } \\ \chi^A\left(((2a + 4)\mu_3b^{2} + (4a + 2)\mu_3)c + 4\mu_3b^{2} + 4a\cdot \mu_3 + 1 \right) &= i^{ 0 } \\ \chi^A\left((((2a - 3)\mu_3 + (3a - 1))b^{2} + (2\mu_3 + (2a - 3))b + ((-a + 2)\mu_3 + (a + 2)))c + ((3a - 2)\mu_3 + (a + 4))b^{2} + ((2a - 1)\mu_3 + (a - 3))b + (2a + 4)\mu_3 - 2a + 1 \right) &= i^{ 0 } \\ \chi^A\left(((2a + 1)b^{2} + (2a + 2)b)\cdot c + (2a + 4)b^{2} + 2a\cdot b + 1 \right) &= i^{ 0 } \\ \chi^A\left((((3a + 2)\mu_3 + (2a + 3))b^{2} + (a - 3)\mu_3b + ((2a - 2)\mu_3 - 2))c + (3\mu_3 + (a - 2))b^{2} + (-\mu_3 + (a - 1))b + (2a - 2)\mu_3 + a + 1 \right) &= i^{ 0 } \\ \chi^A\left((((2a + 2)\mu_3 + 2)b^{2} + (a\cdot \mu_3 + a)b + ((-2a + 4)\mu_3 + (3a + 4)))c + (3a\cdot \mu_3 + (a - 2))b^{2} + (3\mu_3 + 3)b + (-2a - 2)\mu_3 - 2a + 1 \right) &= i^{ 2 } \\ \chi^A\left(((4\mu_3 + 4)b^{2} + ((a - 2)\mu_3 + (2a - 2))b + ((4a - 2)\mu_3 + (4a - 2)))c + ((3a + 4)\mu_3 + (3a + 4))b^{2} + (-3\mu_3 + (2a + 1))b - 2\mu_3 - 1 \right) &= i^{ 2 } \\ \chi^A\left((((2a + 2)\mu_3 + (2a + 4))b^{2} + ((2a - 2)\mu_3 + (2a + 2))b + (4\mu_3 - 2a - 2))c + ((3a + 1)\mu_3 + 2)b^{2} + ((3a - 3)\mu_3 + (3a - 3))b + 4\mu_3 - 3a + 1 \right) &= i^{ 0 } \\ \chi^A\left(((-2\mu_3 + 4)b^{2} + ((3a + 2)\mu_3 + 4)b + (a - 2)\mu_3)c + a\cdot \mu_3b^{2} + (3a + 3)\mu_3b + (3a + 3)\mu_3 + 4a \right) &= i^{ 0 } \\ \chi^A\left(((4\mu_3 + 2)b^{2} + ((3a + 4)\mu_3 + (3a + 4))b + ((4a + 4)\mu_3 - 3a - 2))c + ((2a + 1)\mu_3 + (3a + 2))b^{2} + ((3a - 1)\mu_3 + (3a - 1))b - 2\mu_3 - a - 1 \right) &= i^{ 0 } \\ \chi^A\left((-\mu_3 + 3)c + -\mu_3 + 3 \right) &= i^{ 0 } \\ \chi^A\left((((3a + 2)\mu_3 + 2a)\cdot b^{2} + ((a + 4)\mu_3 + (3a - 2))b + (a - 2))c + ((2a - 1)\mu_3 + (3a + 2))b^{2} + ((a - 1)\mu_3 + (3a - 1))b + (-2a - 2)\mu_3 - 3a - 1 \right) &= i^{ 2 } \\ \chi^A\left(((2a + 1)\mu_3b^{2} + (2a + 2)\mu_3b)\cdot c + (2a + 4)\mu_3b^{2} + 2a\cdot \mu_3b + 1 \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} + 51831712670854041786239427056a - 579427966205952858256958650032 x^{47} + -230463722507799963773268200156a - 490105524727471407652038357852 x^{46} + 361488191718384518569518174324a - 114528293154892907441781366880 x^{45} + -380170267204248102946062614508a - 8233869634088426349317765356 x^{44} + (-77179833850650988702292300788a + 526211696476074180737318326708 )x^{43} + (-32202480136899191481039631700a + 532914766139064576539500874864 )x^{42} + (343013233199559777342950047856a + 201931237180398795030423972728 )x^{41} + (-566827457610040634553973386620a + 178876552308010864501468566336 )x^{40} + (459386443466936412012061199968a + 333425852940284400798754566816 )x^{39} + -230647399298957287892794000516a - 123809707607455479400227285716 x^{38} + (-378486829267825920753713490472a + 403424068814696332553161470828 )x^{37} + -107453290984239646909996799228a - 185594615318430233762710545864 x^{36} + 80216497592104752343808422744a - 153128142886111031762409810776 x^{35} + (-311955441992156237288071160252a + 482804772226251624909878472288 )x^{34} + (-142340936922933993627400141508a + 135126883891792049576256866520 )x^{33} + (-577414295357337127937344320428a + 97060677393863436760828590248 )x^{32} + (-313401411454345177224009551828a + 385102829594314075655856055744 )x^{31} + -589637833900846193662994603908a - 159454467720910363127525276612 x^{30} + -341889013329552000868038386892a - 519570907522904496380511528736 x^{29} + (-108167873497240351592842952952a + 13262180476023809120514208792 )x^{28} + -92549956292424245074889199240a - 357916786388688553262318319960 x^{27} + -507655224844352585322017780776a - 553421821919401897720797482204 x^{26} + 789294429847390336132090940a - 559086914460837330465328927208 x^{25} + -415157613056276627939447435518a - 284028841865777138566783696596 x^{24} + 209930465557563406235399450576a - 373704280496057712898851989424 x^{23} + 554153539818093924710619904932a - 34499661590083448624098356032 x^{22} + (453996873495483240250160632440a + 290669807312898096349893152536 )x^{21} + (238332977928900190979727402756a + 511727426194368544910606007272 )x^{20} + (48652799512869818677880233828a + 39857150274567496849521827288 )x^{19} + (409657429846583121774582744240a + 14989444146831898908142153576 )x^{18} + -599269841204118580320172547536a - 274503504790642498907162052896 x^{17} + (41386380915581224888223685376a + 286486961156243558309741404024 )x^{16} + 338690438001298056647720586776a - 560003565603678711099017810288 x^{15} + 633676834389624894854286534692a - 395198096485681041950902080000 x^{14} + -492227828623968629035972064348a - 314634691956392049241276275688 x^{13} + 280726352440533871671656825624a - 273844725148947373757151669420 x^{12} + -389813224556733536546325828000a - 33946098725025955979561574176 x^{11} + -619391264772206878694425197480a - 147128516536547707702157094248 x^{10} + -103808453005345741786102744192a - 28648902563581976153797075784 x^{9} + (607754624718273240602860773320a + 481618054996810238977050633376 )x^{8} + (-366356894019718872214275630976a + 102975146188771671005531997176 )x^{7} + (484956202743046519003487578156a + 10686365661385664682382560992 )x^{6} + 48854871253656583584998201592a - 458498678256729893266033753064 x^{5} + (-261046438015888301406998981344a + 432720769599023838138372714512 )x^{4} + (-251045071939437681018493763560a + 482352257127093153038561743184 )x^{3} + 91007008110534956615657301116a - 326199352344433567913936639576 x^{2} + (563818158600984405327415392920a + 524216414117727771349074228456 )x + 347660800050225073451377764772a - 280657053886776880327547316658 \)
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