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ex.24.10.1.127_255_383.a

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

Inertia Polynomial

The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + -592169270313875550061727782928a - 127521892742254812429171944152 x^{47} + -380205272365862522776369934692a - 520814197357727158797283199060 x^{46} + (534546328242101487550746799452a + 391485944855345803431133352992 )x^{45} + -221072582608588666794607517484a - 236580671863652841735275365272 x^{44} + (563928818490201410418223680320a + 406245975777262917370655775264 )x^{43} + (389311454493684673253768470564a + 604815304335225720749985240556 )x^{42} + 302475875767100616322451161600a - 548041857453063597153079328696 x^{41} + 125671249607798963924647155956a - 218976963090430693129184973244 x^{40} + 153105617768407250187411618824a - 64409637118288893263945463344 x^{39} + 607392548545804162834873709560a - 189087559257346594979436064432 x^{38} + (-159428422471987241064936326812a + 151373626844602209119129870520 )x^{37} + (540644210196493963842904042764a + 319643822504334119565930314684 )x^{36} + 94187120701606776936735307224a - 485833399555016056951647099328 x^{35} + 480695150298982885526294400616a - 280332409521165556389181642544 x^{34} + -271256747848967672554857600664a - 105337277183521373544597547048 x^{33} + -614187161104276110928377312836a - 318542724926060596956095075632 x^{32} + (429679018697634090036462269312a + 60157836676218662964454485600 )x^{31} + (-221464848457767212752708057616a + 157946045433460870470550732768 )x^{30} + (-395637477142749426259514833976a + 424746042902743734628352060128 )x^{29} + -349062080418526225051395627488a - 476017552663350708359592508136 x^{28} + (-258242275209519519787568158416a + 210826746118040977198730286624 )x^{27} + 435694671424090607553109308344a - 468048779332496828313895328040 x^{26} + (-303893593517578183658139100064a + 277197679567674450688410491032 )x^{25} + -172192121995137333626536636738a - 470605063506196765819605382104 x^{24} + 552324329508076751272438343800a - 199838380878775530579932820928 x^{23} + -333502207976834185725723907596a - 490645692610235566156077162728 x^{22} + (185777758648251363146755287408a + 540079177201516688509978811944 )x^{21} + (-483880428610881642111330274648a + 56660976721336524459090363816 )x^{20} + (123643447316746969241052227824a + 320997614641605967142674331152 )x^{19} + 307421686691277665748246333804a - 140420722324151166004977694264 x^{18} + (-271664406158704931582337090936a + 354723486780688506577618213472 )x^{17} + (102251026308755588829479137372a + 254354438386830934540804240584 )x^{16} + (-95214774780900790631160772112a + 60033957485075917915337683616 )x^{15} + (-75993630153842508338004118720a + 562621574741169791237728848336 )x^{14} + 6559594680111802294713293448a - 219735895155715190022338268744 x^{13} + -571085199272944527333782145800a - 264470490009389175067194010816 x^{12} + (-593475607755643873313055832464a + 476234875244863542509355753520 )x^{11} + -67829420827544995838218609032a - 441028193847012474555574092304 x^{10} + 361926924932465733599999597032a - 228410242429741072429272720784 x^{9} + 155022749497060550716899331152a - 533547256110067846166124908904 x^{8} + -23283093901081851356276600576a - 622427406623683174553597057488 x^{7} + 543879113425272752832209375872a - 311715282760776472130278762144 x^{6} + -104463404821963011845596217328a - 278544042852467694819712645904 x^{5} + (346124216119107818072892828920a + 349524371216516480977906987232 )x^{4} + (-476529063541843568314858211440a + 143864814519213364526630102464 )x^{3} + -149052846516653182680415369048a - 36192374780829686467060265600 x^{2} + (-526060844081989083214442432728a + 427785832406701550341993086112 )x - 563902603590369499222959693952a + 508465315790246401330873207298 \)
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