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ex.24.10.1.31_63_95.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 \(x^{2} - 211275100038038233582783867563b^{3} x + (-211275100038038233582783867563\mu_3 - 211275100038038233582783867563)b \)

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

Inertia Polynomial

The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + (-258463008709453818939828714228a + 17930292236923456782004876992 )x^{47} + (-356046231306765354266357265156a + 158520601509397497376996721392 )x^{46} + (207565281949374011746648696736a + 391910333364590667454508911344 )x^{45} + (371726727430951334818248068888a + 347729588344424822841749591824 )x^{44} + (237182979901101736105747564392a + 388299097409459320189927295156 )x^{43} + -135230374697112544097647746252a - 620348394005098320786239876792 x^{42} + -453826720171929323657170925728a - 203458561168534887968591157528 x^{41} + -140560196625325270172972111488a - 398007304638193541075442380168 x^{40} + (-559400510685599852762708421584a + 156253169054578118079787878384 )x^{39} + (-515445398581192404447140754160a + 281963113397892403261068242372 )x^{38} + (22511376175265088003792621696a + 236407789126496357219678136096 )x^{37} + 113693811314719525784353114464a - 123195934270874191117301942264 x^{36} + 52523948368200233279085410960a - 358181194162011002300060540184 x^{35} + 560921004224681137322198796452a - 41870958607538508297958447216 x^{34} + -517094749825709846297216860392a - 428431138877061566678805960264 x^{33} + (90862087886949665215520143980a + 171371443729104240853630207608 )x^{32} + (611406353468484024259094674032a + 603161466922841413340985232544 )x^{31} + (-405716180138255037973552699648a + 290733583893689089679703867412 )x^{30} + (523641435193896755185038051984a + 485792260539605629788512779872 )x^{29} + (-513396463417329936378489557888a + 150450572005276590183963672160 )x^{28} + 630986314146281178326784354352a - 557182180132085500825672253808 x^{27} + (484049241708201452176211208832a + 177060253009191076017240387400 )x^{26} + (-182400415863416451679728468048a + 285499483165769973105651270104 )x^{25} + -110490681463466434940417265810a - 464074886239488315770896988304 x^{24} + 236033157472262948552819849520a - 182652288973265380771088190440 x^{23} + -122118767342448558218578830528a - 358364217744116830404011523080 x^{22} + -262695382346940842844339589072a - 317731854364451624143942337728 x^{21} + (-202403289429189307601236578408a + 391816452903083419898717977656 )x^{20} + (-323300574094331980507965002660a + 385929512643791543584759855888 )x^{19} + (-506232357401903049823265477512a + 20990108178992158508095703864 )x^{18} + 345270286699379303912104436680a - 339296882787570670451281808096 x^{17} + (-446646269680220896253783997952a + 482881303856350327173895796272 )x^{16} + (-84951487066537411862864729408a + 509256309066087077270387753200 )x^{15} + (260936333720262015425285687444a + 513404915656884311903610456880 )x^{14} + -449940164448138473234908994064a - 486127467736491257008901220512 x^{13} + (-366587016584407828363949546456a + 421524877561038488760789440996 )x^{12} + (-148336115556508619706675219784a + 505607426037075474364636269312 )x^{11} + -94049327257631284132084440608a - 371301862316202089865750458008 x^{10} + -564015541710943491784175165144a - 357810942940946970675052115120 x^{9} + 429216231157200839808183392408a - 429655573310520314688647490056 x^{8} + (-536412023797204688745702313920a + 504183052340527021344783000064 )x^{7} + 167779559553655972230149783460a - 286500782532191129884070723008 x^{6} + (436743913077892227937156308256a + 481900651586348322098565864736 )x^{5} + -210490585038066362449073966048a - 81094938621325280093247482480 x^{4} + (-378872550390823801387063301600a + 409202491848314681395001274304 )x^{3} + 178034186021273624012960305256a - 224667801281047913241043585088 x^{2} + (-64654711155493293130293506480a + 387003866608275749310063043904 )x + 532251549684971087970421223776a + 606220937646742317078318025482 \)
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