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ex.24.8.1.31_63_95.a

Base Field
\(F = \) 2.1.2.2a1.2 \( = \mathbb{Q}_{ 2 }(a) = \mathbb{Q}_{ 2 }[x] / (x^{2} + 2 x + 3\cdot 2 )\) 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
\( v(N) = 8\)
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 \(x^{2} + ((70425033346012744527594622521a - 70425033346012744527594622521)b^{3} )x + ((-140850066692025489055189245042a + 140850066692025489055189245041)\mu_3 - 140850066692025489055189245042a + 140850066692025489055189245041)b \)

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 + ((3a + 1)\mu_3 + (2a - 1))b^{2} + ((3a + 3)\mu_3 + (2a + 2))b + 2a\cdot \mu_3 + 3a - 3 \right) &= i^{ 2 } \\ \chi^A\left((((3a - 2)\mu_3 + (a + 4))b^{2} + (4\mu_3 + 2a)\cdot b + 4a\cdot \mu_3)c + ((2a + 2)\mu_3 + (2a - 2))b^{2} + 4b + (a + 4)\mu_3 + a - 3 \right) &= i^{ 0 } \\ \chi^A\left(((2a + 1)b + (2a - 2)\mu_3)c + (a - 2)b + 1 \right) &= i^{ 0 } \\ \chi^A\left((((2a - 1)\mu_3 + (2a - 1))b + (2a - 2))c + ((3a + 2)\mu_3 + (3a + 2))b + 1 \right) &= i^{ 0 } \\ \chi^A\left((((2a - 3)\mu_3 + (a + 1))b^{2} - 3b + (-a + 3)\mu_3 - a)c + ((a + 3)\mu_3 + 3)b^{2} + ((a + 4)\mu_3 + (2a + 1))b + (3a - 3)\mu_3 + 4a \right) &= i^{ 2 } \\ \chi^A\left((4\mu_3b^{2} + ((2a + 4)\mu_3 + (3a + 4))b - a\cdot \mu_3 + 4a)\cdot c + 4\mu_3b^{2} + (4\mu_3 + (2a - 3))b + (-2a - 3)\mu_3 \right) &= i^{ 0 } \\ \chi^A\left((((3a - 2)\mu_3 + 4)b^{2} + (a - 2)b + ((4a + 2)\mu_3 + (2a - 2)))c + ((a + 1)\mu_3 + (2a - 1))b^{2} + ((3a + 3)\mu_3 + 2)b + 4a\cdot \mu_3 + a - 3 \right) &= i^{ 2 } \\ \chi^A\left((((a + 4)\mu_3 + (a + 4))b^{2} + (2a + 4)b + (-2a - 2)\mu_3)c + (-2\mu_3 - 2)b^{2} + (2a - 1)b + (4a + 3)\mu_3 \right) &= i^{ 0 } \\ \chi^A\left(((2a\cdot \mu_3 + 2a)\cdot b^{2} + (2a + 4)b + (4\mu_3 + (4a + 4)))c + ((2a - 2)\mu_3 + 4)b^{2} + ((2a + 1)\mu_3 + 2a)\cdot b + (2a - 2)\mu_3 - 2a + 3 \right) &= i^{ 2 } \\ \chi^A\left(((4\mu_3 + (2a + 4))b^{2} + ((a + 2)\mu_3 + (3a + 4))b + ((-3a + 4)\mu_3 + 2a))\cdot c + (2a\cdot \mu_3 + 4)b^{2} + (2a + 1)b + \mu_3 + 4 \right) &= i^{ 0 } \\ \chi^A\left(((-\mu_3 - 1)b^{2} + (2a + 2)b + 4\mu_3)c + ((2a + 4)\mu_3 + (2a + 4))b^{2} + 2a\cdot b + 1 \right) &= i^{ 1 } \\ \chi^A\left((((3a + 4)\mu_3 + 3a)\cdot b^{2} + ((2a + 4)\mu_3 + (3a + 4))b + (2a\cdot \mu_3 + 4a))\cdot c + (4\mu_3 + 4)b^{2} + (4\mu_3 + (2a + 3))b + (-2a - 1)\mu_3 + 4a \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 } \end{array} \)

Inertia Polynomial

The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + 490438387088150914502990015284a - 181060457920262755630519004600 x^{47} + (-588581141271029579928069408296a + 525078759220995660387022407880 )x^{46} + (110568725230852461309796310204a + 245491970720374987021251458520 )x^{45} + 533441823382676683538547083064a - 172073729442407149308425988292 x^{44} + (372484984130910894819942204896a + 27545088632196701267767194656 )x^{43} + (-407150843021448228741260536396a + 288980438020767025956015325684 )x^{42} + -203804141528785313698876910644a - 266889346829333569286591038952 x^{41} + (516276899819033403954337586724a + 533546258654414519218315936188 )x^{40} + 140829399274162276675607239680a - 28669587796734228350905283216 x^{39} + (-122550206862704021697732696244a + 579138872267646416832568507780 )x^{38} + 587286413298184990094780951512a - 263489085506776833810737148740 x^{37} + 324058309459765059372585461392a - 364826207663375469830109547748 x^{36} + (467912147159727295185551431984a + 243314174215261179068588008112 )x^{35} + (448939669199404326469144763212a + 603898568513681095254100604904 )x^{34} + 415126512998462082003118341632a - 19710883850422611285554308176 x^{33} + -487475764595479651862497071296a - 433254696198306873437316047728 x^{32} + (371530927985454046388622385384a + 237594429436603717209326523156 )x^{31} + (9094710304165559244288415116a + 502568888903719364191368435644 )x^{30} + (22226491526893863510431399424a + 628451474283185560396731770896 )x^{29} + (93179637626015558429569311756a + 617247773377163658673878413584 )x^{28} + (-234381170931505852094055610832a + 554979894685869012379687280432 )x^{27} + (340465348282073081509044334600a + 284016873397860130667353695848 )x^{26} + 448852427822680278188254989656a - 475026942708409887902818994008 x^{25} + (-220606618335925942125020238190a + 215813502222233146101155484424 )x^{24} + 443784435866624238310626556544a - 174835536016311011699371043720 x^{23} + (-468637305144076760796431283048a + 454999913181393075033582256616 )x^{22} + -297587394930297664261922233280a - 107923629459899431991171996520 x^{21} + (-165095353861782602697901923316a + 370853314391373280822960796816 )x^{20} + (105233837509923979164169998264a + 81506320732289038063692079024 )x^{19} + 7846887480437168077756202140a - 410808437348330503530118446768 x^{18} + (143322362994840183239816440816a + 144532222963942882512301782904 )x^{17} + -432422768113610332018497529692a - 59910340029872887021599956256 x^{16} + -116175964426186944502174577456a - 27021295577148788765483147008 x^{15} + 358685656099119745876468771292a - 125592883003896138545694488208 x^{14} + (459210069657035072976227841468a + 362369147910033580311164524848 )x^{13} + (-31602275528265436487601488276a + 89560893496732672171585536340 )x^{12} + (189760815321031956371698121168a + 282570262391663923214852748192 )x^{11} + (-142725620261415541682522949224a + 351104260993850821924269966632 )x^{10} + (-122653657595394954907069121248a + 596957586789867787325998468912 )x^{9} + 176092293633394046923976047400a - 375218784541241424193178707584 x^{8} + (-625402795220388271671854318900a + 184250187951931734857510465496 )x^{7} + -324443147163915349733418908428a - 632290295896748641667503243352 x^{6} + -72179925623234753603957042640a - 344373411142183443037592728144 x^{5} + 90817624434641810092749551240a - 20956046786697218259009802456 x^{4} + 630340679992825403837224313008a - 342796054579846052018023171504 x^{3} + -72424604508667212691970904872a - 78913823397359708437057322608 x^{2} + (-176423074304218876742063318208a + 560360224115380797141575351888 )x - 610416283558611544420514781910a + 466776003813790824593052750982 \)
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