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ex.24.10.1.127_255_383.d

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 \(x^{2} + ((70425033346012744527594622521a - 70425033346012744527594622521)b^{4} )x + 3b \)

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

Inertia Polynomial

The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + 48827215950084103515872244968a - 622341404757242972621050508504 x^{47} + -530384561009546819198330052388a - 123623223858262628505656067000 x^{46} + -614794144669829117765999612780a - 472322810308173580418399926560 x^{45} + (595023153499317357167919114892a + 539510079530272507437798120012 )x^{44} + (578030220123502007681727538416a + 65223879006949830463625825088 )x^{43} + -526570596486568115975963634432a - 75206571107497565234800894996 x^{42} + 425473887312500498557529884672a - 556248904694821053963114792584 x^{41} + 406215996384287654371082406748a - 165877874483044401667592576676 x^{40} + (-429072347777976148924538412824a + 505160191573220126192230702096 )x^{39} + (368335516716424123811721285232a + 569002111421816768325260304592 )x^{38} + (-8779315962607638007673673772a + 276863334380202970720919904440 )x^{37} + (470028044570022722050589042844a + 222028984709823363798168262804 )x^{36} + -469370719622340291886863014520a - 78996497504828737365276002384 x^{35} + 367057893654494648911052928052a - 160255466621544963154156324056 x^{34} + -88891388043664465904998654072a - 222574180385013856456885469792 x^{33} + (528370525694297674071351946260a + 150264550318251885754757757432 )x^{32} + (291497299543150706739308876768a + 37347538896861846201782088928 )x^{31} + 568750151284857453477884896128a - 147966439729075990010608853824 x^{30} + 628323895904851901096208252152a - 484787454630109391197449322088 x^{29} + -249865629869616637689964483772a - 507424685326467686009173360328 x^{28} + (-321690918742913006248188117728a + 440052430083631913135203033840 )x^{27} + (-347117071787205440261605228012a + 453965756679419641920908028912 )x^{26} + 330047279554427741529406903560a - 277003923793097783615210263488 x^{25} + (-87622902991599403562224824618a + 208426871287952274858997774432 )x^{24} + -620226713767475044553596778520a - 151027746606166757300483804688 x^{23} + (-324942418339693599380117775952a + 426261832062186982993130510552 )x^{22} + (-488411256555815385954192427816a + 247065599950158085493086584504 )x^{21} + 516000576262435133406607991228a - 564303490690414861766241820680 x^{20} + (632855374765724114346120481136a + 95157346007566628205358837488 )x^{19} + -109277273922713964208891223044a - 610886472703458594898511416336 x^{18} + (-287252520973633456477265364584a + 242615299518537846022058053072 )x^{17} + -436458200437732165162164611196a - 307550244443034814350404636200 x^{16} + (426985432597173903946897773216a + 524858901683627280866576756736 )x^{15} + 17569839540412330305288719912a - 334469780008925329853197056512 x^{14} + (3904897962534050040002822208a + 443093871625809557962897143576 )x^{13} + 178149640304915735444126885200a - 227213647756023240862039884592 x^{12} + (43827743999272542117126765296a + 110892406458208511940552165296 )x^{11} + (-87960524260492468704003559256a + 97958579077098239800249323432 )x^{10} + (268306847964804330107597286960a + 184256272133002121351532131248 )x^{9} + (-105214436932930763778914619776a + 494690390237309329678226759064 )x^{8} + (56719930521680067058260055728a + 180435802584711890561485853648 )x^{7} + (-514480650955748212111240542496a + 11563139273919929331936480160 )x^{6} + (-40589368737397211454483433368a + 421476895619568650261571977200 )x^{5} + 551579167942611446238734617104a - 411872247188884215890537670376 x^{4} + (339536893584047468634074018704a + 151363793307420571044767155104 )x^{3} + -34321472108370103536409499000a - 163282384977949949180991158552 x^{2} + -524826602245197903365119778320a - 344475553244588828072281367888 x + 256590456865975041363387682934a - 290998703942033750342412280766 \)
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