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ex.24.10.1.31_63_95.d

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
\( v(N) = 10\)
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 \(x^{2} + ((a + 1)b^{3} )x + (-\mu_3 - 1)b \)

Underlying Character

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

Order
4
Conductor exponent
20
Values on generators of \((\mathcal{O}_K/\mathfrak p^{ 20 })^\times/U_{\mathfrak{p}^{ 20 } }\) :
\(\begin{array}{l} \chi^A\left(\mu_3c + 1 \right) &= i^{ 1 } \\ \chi^A\left(((2a - 2)b^{2} + (-2\mu_3 - 2)b + ((4a + 2)\mu_3 + 4))c + ((a + 1)\mu_3 + (2a + 3))b^{2} + ((a + 1)\mu_3 + (2a - 2))b - 2a\cdot \mu_3 + a + 1 \right) &= i^{ 2 } \\ \chi^A\left((((a - 2)\mu_3 + 3a)\cdot b^{2} + (4\mu_3 + 2a)\cdot b + 4a\cdot \mu_3)c + ((2a - 2)\mu_3 + (2a + 2))b^{2} + 4b - a\cdot \mu_3 - a + 1 \right) &= i^{ 0 } \\ \chi^A\left(((\mu_3 + 1)b + (2a + 2))c + ((3a - 2)\mu_3 + (3a - 2))b + 1 \right) &= i^{ 0 } \\ \chi^A\left((\mu_3b^{2} + ((2a - 2)\mu_3 + (2a - 2))b + 4)c + (2a + 4)\mu_3b^{2} + (2a\cdot \mu_3 + 2a)\cdot b + 1 \right) &= i^{ 0 } \\ \chi^A\left((-b + (2a + 2)\mu_3)c + (a + 2)b + 1 \right) &= i^{ 2 } \\ \chi^A\left((((2a + 4)\mu_3 + 2a)\cdot b^{2} + 2a\cdot b + (4\mu_3 + (4a + 4)))c + (2a + 2)\mu_3b^{2} + (3\mu_3 + (2a + 4))b + (-2a + 2)\mu_3 - 2a - 1 \right) &= i^{ 2 } \\ \chi^A\left((((a - 2)\mu_3 + (2a + 2))b^{2} + ((3a + 4)\mu_3 + 3a)\cdot b + (3a + 2))c + ((a + 3)\mu_3 + (3a - 3))b^{2} + ((3a + 3)\mu_3 - 2)b + (4a + 2)\mu_3 + a + 3 \right) &= i^{ 2 } \\ \chi^A\left(((3a\cdot \mu_3 + 3a)\cdot b^{2} + (2a + 4)b + (-2a + 2)\mu_3)c + (2\mu_3 + 2)b^{2} - 3b - \mu_3 \right) &= i^{ 0 } \\ \chi^A\left((((2a + 4)\mu_3 + 2a)\cdot b^{2} + ((2a + 4)\mu_3 + (3a + 4))b + (3a + 4)\mu_3)c + 4b^{2} + (4\mu_3 - 1)b + (2a + 1)\mu_3 \right) &= i^{ 0 } \\ \chi^A\left(((3a\cdot \mu_3 + (a + 2))b^{2} + ((a - 2)\mu_3 + 2a)\cdot b + ((a + 2)\mu_3 + (3a + 2)))c + ((2a + 2)\mu_3 + (a - 3))b^{2} + ((a + 1)\mu_3 + 2a)\cdot b + (a - 3)\mu_3 - 3a - 3 \right) &= i^{ 0 } \\ \chi^A\left((-\mu_3 + 3)c + -\mu_3 + 3 \right) &= i^{ 0 } \\ \chi^A\left(((a\cdot \mu_3 + (a + 4))b^{2} + ((2a + 4)\mu_3 + (3a + 4))b + 2a\cdot \mu_3)c + (4\mu_3 + 4)b^{2} + (4\mu_3 + 1)b + (2a + 3)\mu_3 + 4a \right) &= i^{ 0 } \\ \chi^A\left(((-\mu_3 + 3)b^{2} + (3a\cdot \mu_3 + (a - 2))b + (4a\cdot \mu_3 - a + 4))c + (-3\mu_3 - 3)b^{2} + ((a - 2)\mu_3 + (2a - 2))b + (2a - 2)\mu_3 + 4a + 1 \right) &= i^{ 1 } \end{array} \)

Inertia Polynomial

The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + (-503805321684194532527988369556a + 443308299668500445543355558952 )x^{47} + (-516606601694018118127998629028a + 441090585327384812851934563192 )x^{46} + (-201391684031019385461746366760a + 552313628548831270684533791448 )x^{45} + 477963782157761030069076847180a - 483248425125943073759325443240 x^{44} + -296010123817020231454404813928a - 609732982633670902374880794284 x^{43} + (542547105240973871277474998756a + 393251874311284921800209211320 )x^{42} + 323589194550341789207479379848a - 196272914339021746548007392384 x^{41} + -82475203077541539554896283400a - 365589988883019336991792961464 x^{40} + (295806169308979557627269683768a + 147678848444145698712738091952 )x^{39} + (620261205973624530674325307840a + 604882055231793218814740376140 )x^{38} + 532291048742117470561443126024a - 566986755220799540211426735216 x^{37} + (-451594873909407094716460302968a + 582593074041454420881305819800 )x^{36} + -440608710722001892019811117808a - 159968327345330004032614002320 x^{35} + (-413056506691836236689431895880a + 93612498775369700548642482048 )x^{34} + (11719142386000125813573751320a + 279292817430292569036490837112 )x^{33} + 419919518145607175301228156048a - 157657102959781074381406596720 x^{32} + (479667842651839581148998767888a + 354186008336179784123992631872 )x^{31} + (-308857262186049906925602100932a + 621865566905554486216861829260 )x^{30} + (190633891423560568016346777884a + 50843056410373930609724387856 )x^{29} + (147517304705925007469919280208a + 86610214160640535476810595080 )x^{28} + (-393612859376353437981532672272a + 129711872267314303619736003312 )x^{27} + 234440245793182351277513501592a - 508013859521435637966455337784 x^{26} + -175121328213931451580321595888a - 632176642381464828999856437840 x^{25} + 250474861838802663047024147142a - 148324628129775085995894953712 x^{24} + (-36322859701006549082010728624a + 45521278172341608041521639880 )x^{23} + (-414858598379380109957056824688a + 393700925856394717731567911624 )x^{22} + 349561141552657250239302828648a - 432860811545872398857074086832 x^{21} + 220301326142046439664441376944a - 435986240070810516510391725408 x^{20} + (-155782473927942783642405982324a + 296432414799316288950650537424 )x^{19} + -216250894743100353645533737616a - 154524791021249572887653835896 x^{18} + (-506352187759308530027227768208a + 251156003497122904082592842976 )x^{17} + -539011508297353257117233920296a - 566926045072478671645181953728 x^{16} + (455590606116576752844359824832a + 566998608305702999357188171008 )x^{15} + (68263031466727059776044408964a + 489002750898442459316333849760 )x^{14} + 52995423521730433209353688752a - 259604607318384266714153711984 x^{13} + 565509230515305967334517318072a - 147975886368850668055003723124 x^{12} + (15427414179410921589881030488a + 211102700866590082866491988656 )x^{11} + 17632314349010814097444307712a - 84239170999674153353745159056 x^{10} + -215187502751556036867735953256a - 99595401842782519298560267600 x^{9} + (435312196999538003474362547120a + 600936278560845154347387263024 )x^{8} + (554093965408942958769212471456a + 293427882514504018446248347824 )x^{7} + 611101571380567506081468705508a - 35294136116622262077557939416 x^{6} + (-60856729630845977855069057464a + 265759293650277626173226575736 )x^{5} + 276087514464384864486077652168a - 38603724194999961995624318768 x^{4} + 581597213157893495145706371488a - 41529004255788573524994029152 x^{3} + (-413794895287421806160999869304a + 482449318697590170976294188224 )x^{2} + 197803710155894842696513455336a - 630705996714355675535627314256 x + 229324656513965904187963410414a - 114429724750077240677276545954 \)
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