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ex.24.10.1.127_255_383.b

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 \(x^{2} - 211275100038038233582783867563b^{4} x + b \)

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

Inertia Polynomial

The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + -294752229081738125607234122768a - 528931595222258154223109781144 x^{47} + (-464049770391184870334838073464a + 350839200378232095374433807524 )x^{46} + (-92062300096016363777636823396a + 270232756684474359204725133488 )x^{45} + -594212266976526147343218784232a - 583502706192966509238966963696 x^{44} + 501309416210798575613642820096a - 321302324119066913165737234640 x^{43} + -206902379643241269792525916788a - 133634440021570219900036924644 x^{42} + -441444777711752973294799463360a - 128093673248113985549160673464 x^{41} + (324744692798911823664015988360a + 358364416384417311748295343212 )x^{40} + -568543848123392056907741609816a - 283188422655623500245589520096 x^{39} + (-133597299736666091406345744896a + 326204908713970927405826592280 )x^{38} + -409717335919255205495242503476a - 472772182061202799625174872048 x^{37} + -358523437238637517847100440120a - 168123795056244558703752455524 x^{36} + -77830220093346650686763474072a - 52715764636626370072072206880 x^{35} + 241775324582995764411742682232a - 471939831597261359288292282456 x^{34} + (-601001948670131544220928702680a + 70531950821758011686204528096 )x^{33} + (165930102743743479586276390688a + 259228998178691919416345035952 )x^{32} + (-293627676243096556672662680576a + 518009460616536089192371577088 )x^{31} + (-405846034624674903295576145928a + 59963185435989951560447100128 )x^{30} + -23201454751369369328252477544a - 153543766535661171916454403344 x^{29} + 343610422950658242947864688252a - 280183673206558907384882206816 x^{28} + (-365740722073225425407775361760a + 56044546839846365717759200432 )x^{27} + -246541900016225998134304020648a - 367177084021398706518203036264 x^{26} + (209587883966959049618230647496a + 349860247375500483235223018216 )x^{25} + 139881177566113085337820932702a - 424742378020969381989694560200 x^{24} + 10701254918749334649240704088a - 180878737916177911184360607328 x^{23} + (38479295369286491148254960020a + 358480914792660737256120619728 )x^{22} + (-144763601300727931407753444048a + 480029770082979733468342539464 )x^{21} + -127999681311835559121209448032a - 531200426526252508353405707584 x^{20} + (-592774444258745629527692711248a + 30112167396177104883313073936 )x^{19} + 474245676678415921510434953612a - 460495593008247074000368780328 x^{18} + -217371305614838262258152405608a - 300184724161757377023239183776 x^{17} + (-147365521364842798844919062780a + 181175197969809453857064331984 )x^{16} + (69247968619974621344177216096a + 273106625914457520837034356768 )x^{15} + (25974761308658603089611131160a + 203404165990062121214455889344 )x^{14} + -488391916985426441626939751536a - 532757004808766215191819837272 x^{13} + 49063374142270169542598439896a - 555026764463665504432342006568 x^{12} + 298157497219475210628806680496a - 11575328824847263706943835024 x^{11} + 308794123948073463212909834448a - 543864861603635592615743009952 x^{10} + (-504385146338801878002829616272a + 173442996778553515653474970032 )x^{9} + (266546213313361680642091214752a + 287063352628138391539748310400 )x^{8} + 10119220484676841177862870944a - 45308451560665376059217532240 x^{7} + -438704220446134306852579531520a - 468964574659290602356478332608 x^{6} + (-275314300458888673559781886480a + 304539948800620386675531610928 )x^{5} + -48084626408174250836343320176a - 386224909122262823764661130808 x^{4} + -198222378160394101660238938832a - 414502576219765688132236406240 x^{3} + 230544781891617161749917853688a - 563842986763653372421016550480 x^{2} + 240881864416275329395103311048a - 502088649800446466914805309488 x - 183129658291636843687062514984a + 572491384927252902355730455602 \)
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