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ex.24.10.1.127_255_383.a

Base Field
\(F = \) 2.1.2.3a1.2 \( = \mathbb{Q}_{ 2 }(a) = \mathbb{Q}_{ 2 }[x] / (x^{2} + 5\cdot 2 )\) 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 \(x^{2} + 253530120045645880299340641075b^{4} x + b \)

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

Inertia Polynomial

The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + -148448310797480409538846019856a - 250306187844320568896208682088 x^{47} + (-403786074718168736588881675668a + 255819163758863282838702936784 )x^{46} + -103229726520991387046263671204a - 624838377020949983982936785600 x^{45} + (340481003733433057302370056652a + 548446631608783823007954190564 )x^{44} + (-496173437699025322015492125408a + 190237026178719687869431760608 )x^{43} + 282801337809207829559866592900a - 566228710510772746870841786900 x^{42} + (-491247014236987596337734669752a + 528411597556070492758003163880 )x^{41} + 223339739796048020716389602344a - 216775534503720777129577362908 x^{40} + 314010159765163906688907221496a - 360089075452960232241518606128 x^{39} + (-382877017234124048926760706468a + 425963185608829776771259375520 )x^{38} + 294625531003873511052214003172a - 389486439219505905719669632016 x^{37} + 164579939795315785168563706864a - 590593006331247291863442290788 x^{36} + 257215893037547469408192512712a - 169821977066328911907822494864 x^{35} + -60780627203553701506030452452a - 98527370811612201932761869648 x^{34} + (-214056744976680113069697948776a + 319295103792865760915668868680 )x^{33} + (174038336677032395404442772112a + 268807729913786458871047765160 )x^{32} + (525879646999396332537919967008a + 541265722874521649594150821568 )x^{31} + -323028958227594628856031456392a - 11843537063976015856081936848 x^{30} + (-214554887310546381517740070504a + 309854184484399420045824856200 )x^{29} + (3396326977733302327715179264a + 442467177406604951738356387616 )x^{28} + (112148771055925860490330854512a + 115228732901262517337495938592 )x^{27} + (-121878050566207655198090990588a + 97776047145985913598209714824 )x^{26} + -225386502253059838917994934960a - 73337405676962899400681360464 x^{25} + 443677081489593045750370176558a - 582913415118604413083374622896 x^{24} + 111813187387264217144013499656a - 72665855078727747669955831392 x^{23} + -71367689061683385246283375144a - 10955732163692738351038048008 x^{22} + 234334792603819006355330936912a - 39648846306974755525034471496 x^{21} + -437642712361403506543685023524a - 253915785910766815207385917720 x^{20} + -95213239288665560292643066608a - 5652889650313884556941012016 x^{19} + -204429103195637604600619128180a - 619310238629450868716261093144 x^{18} + (349508810993310388173374763496a + 180116120613645614895576121760 )x^{17} + (-323542719379411724720180121500a + 259141229909350540867549973152 )x^{16} + -53292355395568477649267542896a - 318534602128165587918159006464 x^{15} + (-365660946807418823337058930136a + 577493420473096010157828833752 )x^{14} + -475879743571215027971515320016a - 412459649724714849735974715032 x^{13} + 101621452458946106305394263552a - 10142839577142558215894345272 x^{12} + -69539437180874355577358667232a - 446152920978076878465581742256 x^{11} + -527459428591291228047424399240a - 258473582755642914836934101160 x^{10} + (379974761562837351238037607208a + 300567276527240182019913215248 )x^{9} + (-348945356916202321072111825432a + 455559643510107767133556780016 )x^{8} + (-543072424592725925076876439104a + 448898473226746469586361927696 )x^{7} + -462601244733581855349956060752a - 44954488047998830572856878848 x^{6} + (254361182652469527170636716296a + 393052678293192675777394654288 )x^{5} + -625340607386530300737744583776a - 553713342751284159803618196000 x^{4} + -587269965117149233160153720688a - 29995505473911255130857966272 x^{3} + (154557379906540280311226744424a + 150841605572317523865882850440 )x^{2} + 267009963916268858400730329696a - 399085612538542820777910065280 x - 580807625108813925927349226392a + 474690320908092794062973539310 \)
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