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ex.24.10.1.33_67_101.a

Base Field
\(F = \) 2.1.2.3a1.1 \( = \mathbb{Q}_{ 2 }(a) = \mathbb{Q}_{ 2 }[x] / (x^{2} + 2 )\) 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 \(x^{2} - b^{3} x + (a\cdot b - 1)b \)

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^{ 0 } \\ \chi^A\left(((2a\cdot \mu_3 - 2)b^{2} + 2\mu_3b + 4)c + ((2a + 4)\mu_3 + (2a + 2))b^{2} + ((2a - 1)\mu_3 + (2a - 1))b + 4\mu_3 + 4a - 1 \right) &= i^{ 0 } \\ \chi^A\left(((2a + 4)\mu_3b^{2} + (4a - 2)\mu_3)c + 4\mu_3b^{2} + 4a\cdot \mu_3 + 1 \right) &= i^{ 0 } \\ \chi^A\left((((a + 2)\mu_3 + (2a - 1))b^{2} + (a + 3)\mu_3b + ((-2a - 2)\mu_3 + (4a + 2)))c + (3\mu_3 + (3a - 2))b^{2} + (-3\mu_3 + (a - 3))b + (2a + 2)\mu_3 - a + 1 \right) &= i^{ 2 } \\ \chi^A\left(((2a + 1)b^{2} + (2a + 2)b)\cdot c + (2a + 4)b^{2} + 2a\cdot b + 1 \right) &= i^{ 0 } \\ \chi^A\left((((3a - 1)\mu_3 + (3a - 2))b^{2} + (3a - 1)\mu_3b + ((2a - 3)\mu_3 + (a + 4)))c + ((2a + 3)\mu_3 + (3a + 4))b^{2} + ((2a - 1)\mu_3 + 3a)\cdot b + 3\mu_3 + 4 \right) &= i^{ 0 } \\ \chi^A\left(((-2\mu_3 + (2a - 2))b^{2} + ((a + 4)\mu_3 + 3a)\cdot b + ((-2a + 4)\mu_3 + (3a + 4)))c + ((a + 4)\mu_3 + (3a + 2))b^{2} + (-3\mu_3 - 3)b + (2a + 2)\mu_3 + 2a - 3 \right) &= i^{ 2 } \\ \chi^A\left(((3a\cdot \mu_3 + 4)b^{2} + (-2\mu_3 + (3a - 2))b + ((3a + 4)\mu_3 + (2a - 2)))c + ((3a + 2)\mu_3 + (2a + 4))b^{2} + ((3a - 3)\mu_3 + 4)b + (-a + 3)\mu_3 + 4a \right) &= i^{ 2 } \\ \chi^A\left((((a + 4)\mu_3 + (2a + 2))b^{2} + (2a\cdot \mu_3 + 4)b + (2a\cdot \mu_3 + 4a))\cdot c + (2\mu_3 + 2a)\cdot b^{2} + (-2\mu_3 + 2)b + (-a - 2)\mu_3 - 3a - 3 \right) &= i^{ 2 } \\ \chi^A\left((2\mu_3b^{2} + ((3a - 2)\mu_3 + 4)b + ((-3a - 2)\mu_3 + (4a + 4)))c + (3a\cdot \mu_3 + 4)b^{2} + ((3a - 3)\mu_3 + 4)b + (a + 3)\mu_3 \right) &= i^{ 0 } \\ \chi^A\left(((2a\cdot \mu_3 + (2a + 4))b^{2} + ((a + 4)\mu_3 + (2a + 4))b + 4)c + (a - 3)\mu_3b^{2} + ((a + 3)\mu_3 + (3a - 1))b + (4a + 4)\mu_3 + 3a + 1 \right) &= i^{ 0 } \\ \chi^A\left((((2a + 2)\mu_3 + (3a + 2))b^{2} + ((a + 4)\mu_3 + (2a + 1))b + (2a\cdot \mu_3 + (3a + 4)))c + (2a - 1)\mu_3b^{2} + (a\cdot \mu_3 + 2)b + (4a - 2)\mu_3 + 4a - 3 \right) &= i^{ 0 } \\ \chi^A\left((((3a + 4)\mu_3 + (3a + 4))b + ((-3a - 2)\mu_3 - 3a - 2))c + ((a - 1)\mu_3 + (a - 1))b^{2} + (3\mu_3 + 3)b + (4a - 1)\mu_3 + 4a - 1 \right) &= i^{ 0 } \\ \chi^A\left(((2a + 1)\mu_3b^{2} + (2a + 2)\mu_3b)\cdot c + (2a + 4)\mu_3b^{2} + 2a\cdot \mu_3b + 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} + (-200685740940763987135427008216a + 444980641746562530360638118632 )x^{47} + 413784270125925069539619811728a - 449531534616418979833940121180 x^{46} + 400593283409565321512817199012a - 580209545886346724532310287856 x^{45} + (-461771032183224182444971334972a + 411052386932172494118818360748 )x^{44} + 56412020733614813903300763916a - 58845537158642762339347644444 x^{43} + (-97150268928834409832718488520a + 182922075320683813166891773488 )x^{42} + 441561836843543730310625307852a - 101741851015577191823818352312 x^{41} + (-168197918489457714497866286912a + 626492261736555851132932989240 )x^{40} + 198697568550911456329734872776a - 187901124227260928092372321456 x^{39} + (-425967700228877090265355639188a + 451758689192826958353897988660 )x^{38} + -116103385481377866452103919260a - 624953277753546431642171262236 x^{37} + -461881723969181728227363201036a - 75567783477389236360934154048 x^{36} + 627328614303885673155856760840a - 25471511053216675741207143488 x^{35} + 495271445202320363869522574952a - 76537068927145643499616791160 x^{34} + 316482039969599641109247525916a - 17063031676240616187782057128 x^{33} + (339814949818873594184574062592a + 222688181847970626564337951480 )x^{32} + (194024981689273164829671572724a + 454902062597736085300323339520 )x^{31} + 394348887863278043415737427396a - 557182948140296213347068484076 x^{30} + 370721036031254820411714485368a - 298444250504014210612325194464 x^{29} + -244225197886787925144648777184a - 198721611134847465308254210264 x^{28} + (-139048384224056724504731296632a + 61173458063307659181578829576 )x^{27} + -613896067535217315211011352736a - 376971234078616427936944718756 x^{26} + (-595754586284791489296805671852a + 363853860698721792087378216704 )x^{25} + (-452567179757551735969689318678a + 496948325010292451725221058924 )x^{24} + (-504301649232849490293739114720a + 161472359474888256712667776256 )x^{23} + (310450292563392359961126611844a + 143686609340970747645447221656 )x^{22} + -442570538334540236882579662264a - 140502725436083564698813849720 x^{21} + (255761401105636686752047715604a + 383453881478373139586585410984 )x^{20} + 524773011705746096450149954636a - 198516989636895544583984827896 x^{19} + -129648042573676123059660377976a - 439892412627484220411001668272 x^{18} + (507755854068392679297552855400a + 12586611432530741678191176264 )x^{17} + -448686165570584702567511476168a - 313825794797654886104478303056 x^{16} + (142733863723568584552693934328a + 587079898882308004524626328736 )x^{15} + (253355968813944645826958265188a + 543883347152866564885656935680 )x^{14} + (195804904764677175236615908100a + 295077870344519629205779150336 )x^{13} + 464473308306802978362814723896a - 91496313111619121568755730060 x^{12} + (185399219381154199391297129104a + 317383926763520554691068725872 )x^{11} + (-163356509655817975337194092232a + 389179585329588287627102076048 )x^{10} + 605365107977171689847015841120a - 551408232594009194757095203256 x^{9} + 521010040369889634364690604912a - 15025003785641794090458747064 x^{8} + (26876838504225384600893017872a + 417094558682328123901595507096 )x^{7} + (-17442217754899122244871981756a + 16059991436293060104178027792 )x^{6} + 219998286998140476570548000560a - 607524744352161302797580628064 x^{5} + 538972806745663311562953904640a - 266872177362306910299184956912 x^{4} + (105787334676841577916781992024a + 303086387794437597084560535888 )x^{3} + (-129222251329632860575014197116a + 336557086012828943775361128648 )x^{2} + 624222743894958379588488569296a - 608116382970771124310591178568 x + 350259036631185853866176743972a - 546913560819489033401633443438 \)
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