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ex.24.10.1.131_259_387.b

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
\( v(N) = 10\)
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 \(x^{2} + ((70425033346012744527594622521a - 70425033346012744527594622521)b^{4} )x + b \)

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

Inertia Polynomial

The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + 105705502971403697356376867080a - 74728410457774133010420432664 x^{47} + 216403588027805855137896542524a - 142562393244973633473612821304 x^{46} + (147867367774195036250863809276a + 124476284294411158646814314688 )x^{45} + -67217672615619684748380376056a - 241030032643194368578013541212 x^{44} + (278419203494177447208417392280a + 514613351883366865409322159744 )x^{43} + -553823567100689334120219563928a - 294211673296863218324934270452 x^{42} + -538701878923356861920430916456a - 569355416168852057759218941304 x^{41} + 278130162250082992170089332424a - 127310095338336952687210455212 x^{40} + -212145018037633545069526135864a - 67350082496246064351306456160 x^{39} + 369416149444844939453384953584a - 428216388525993687666840912360 x^{38} + 260127581719303807473501741964a - 543248098884816809638178627008 x^{37} + (583888687611879302949074393724a + 302748162957609976670371517196 )x^{36} + (-241176798573815595279362209952a + 316148840521694268699875971808 )x^{35} + 122572909943643779136119365700a - 86107843518943567060188118384 x^{34} + (-497659852122195969835971962200a + 499146878655915731568477514832 )x^{33} + (170626084620050842521893089672a + 33090192767559562941255004128 )x^{32} + (-239900578110905949714636995680a + 356610425399424509592362974528 )x^{31} + 34636032518417506249576027360a - 410072931816632199083236973920 x^{30} + (-263981489120261777787590725928a + 255280745313595119178746765336 )x^{29} + (191971302973796929617884028928a + 577261420355971259832014908808 )x^{28} + 68693658214118200822391763712a - 605610460664970806266427973136 x^{27} + (127826945517715193202930302956a + 101582781133539548124808282680 )x^{26} + 407401507998887714538253318336a - 528811385981899521268936246616 x^{25} + (175684257154440677775621860302a + 327618143999953399473829469100 )x^{24} + (-184472076421987347600588334712a + 173766112806898829210864100448 )x^{23} + 341226114237702763307594781328a - 198841980631601946594976075720 x^{22} + 562932983311124089325816144448a - 202553597111598963259089757400 x^{21} + -614657225308819890996989741844a - 400488497783251689688236750024 x^{20} + -628114304593915056649254805872a - 90521785679933873081695884400 x^{19} + (364573388503856393158049023996a + 157041803358445306633088183176 )x^{18} + 339325104781472589977910177944a - 462092146831927619860009775600 x^{17} + (-194546573334901988561676681420a + 248628826653904708621040211256 )x^{16} + (456462590035020051364140609648a + 37757514772657780642931034112 )x^{15} + -165460158361643647355816933968a - 77669414456237136806146681888 x^{14} + (116756385607401733840629811008a + 294150210606111434580378790376 )x^{13} + 587312531046070094107042381192a - 198051344025253136784443687232 x^{12} + 628440109710091351705768947504a - 530514383102395498045038324320 x^{11} + (207089779374151919960457824568a + 339335149553390948866405407048 )x^{10} + (-308366499314103041353778224336a + 379626741171411967670884578960 )x^{9} + -354391941498928664881244337024a - 434345205633624818768216349296 x^{8} + (-334967992726328612985617778832a + 411440685398639904736182882000 )x^{7} + -387037919083530476009082650848a - 75675028635252882081155578240 x^{6} + (-508760935161502334565651824632a + 551268851667657359067091450112 )x^{5} + 607586518134376880528014867864a - 112931536524990145731498531184 x^{4} + (-612527437467301928356519799056a + 485358373423836135279971003328 )x^{3} + (-281436251671945790618976781720a + 448522103681166737726341566744 )x^{2} + (455104605439342705006233597672a + 514096428419473991110507803824 )x + 118232070266136635021224420562a + 12393809721419815315155114726 \)
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