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ex.24.10.1.127_255_383.b

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

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} + (174366936692055641719553589436a + 66440659106048228958707298968 )x^{46} + (-109493976631436789945502835308a + 524785056547335164336748608928 )x^{45} + (286238586671838306318153015524a + 27929264999689450572112256940 )x^{44} + (8326983818592372583158198000a + 72117047901516500422913821712 )x^{43} + -347330399379229996313828541520a - 203011618772894089980247235444 x^{42} + (-257214626267034383760816121728a + 525911196120571331461921486968 )x^{41} + (202218480852435656513036265596a + 29691748099646263601932045316 )x^{40} + (514135829146611020543305138152a + 209319798454226041465971564688 )x^{39} + -335960659517217211526517307808a - 616148631686692056063784023920 x^{38} + 202084362374027506707212877572a - 157683498047913613207464315064 x^{37} + -156936558268512762375540395724a - 245332010164655357058183527820 x^{36} + 38644453867280271431574094344a - 36148184091072914718262036656 x^{35} + 539588012036015233988276456756a - 325766723095103612714189324680 x^{34} + (238197286769531348048391427560a + 185264011259602456495863761584 )x^{33} + (545611030026865852097033714380a + 14365339071679091889844712560 )x^{32} + (116804503023848229228031498400a + 548964392946642310270190651680 )x^{31} + (-512738755235144887269666191472a + 37834905636119242010380179728 )x^{30} + (-349308861126132528688886215448a + 153840774605177590321669747624 )x^{29} + (574113533843673911180014158348a + 42050391719140783159642818680 )x^{28} + (166940460526539338879320313456a + 133393353796325981922443333584 )x^{27} + (186835946316555370345616616324a + 620572600592166287456296590736 )x^{26} + 591907000246490583983045162776a - 158392991256596156121997545904 x^{25} + (600885936898948958275865169790a + 113429756573625656620240201512 )x^{24} + (48591880399560978951643278248a + 173708157883367527419515412272 )x^{23} + (-207588318515071346986483728928a + 416400277096898082293178825720 )x^{22} + (-413722053240765290698483134904a + 350261646234560506386115478968 )x^{21} + -426183692243617030530514649092a - 565487817359547142410207589224 x^{20} + 325896212858043860009441190304a - 294732453778607177682162254000 x^{19} + (-77424614187042109021649816996a + 393923495391289651389678819776 )x^{18} + 605023666393206430674245856024a - 230785227251761522873171876112 x^{17} + (572617350539429212109990844636a + 98955587043150907920739274824 )x^{16} + (43245134606361744011559431584a + 619836594025128219339248697088 )x^{15} + 512635958426804543646000305960a - 597678147545516091120087440352 x^{14} + (195242017754059933017393545856a + 422149099022529824625054953432 )x^{13} + -137229644827125574409579155648a - 25927274252143204339240530400 x^{12} + -118503387036788268922146662384a - 495074587822115910769952728976 x^{11} + (-519024692580252273939755843496a + 187979381712950186317009894312 )x^{10} + -340878835022562836201333360416a - 318743704162875809825855268688 x^{9} + (188424486823009196097037064968a + 161635622542424756787120428872 )x^{8} + -406936000653757121475412903824a - 325649056184597968454290654000 x^{7} + (-533912006079316656007934703664a + 304095852312958302923219121120 )x^{6} + (528824956119781959875555080984a + 460470324721651333091549195792 )x^{5} + (-46042984633495634984190268032a + 577942588484140032811539883976 )x^{4} + (115863231882136507308826220592a + 466046144995496182728713699264 )x^{3} + -619152689198787513693333299608a - 545553622546633033588576327352 x^{2} + (508804703208689123545990170976a + 609764105724740725444024420336 )x - 386708247107840205144466118770a - 255275795019750265346874107198 \)
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