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ex.24.10.1.33_67_101.b

Base Field
\(F = \) 2.1.2.3a1.3 \( = \mathbb{Q}_{ 2 }(a) = \mathbb{Q}_{ 2 }[x] / (x^{2} + 7\cdot 2 )\) 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 \(x^{2} + 90546471444873528678335943241b^{3} x + (-90546471444873528678335943241a\cdot b + 90546471444873528678335943241)b \)

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\cdot \mu_3 + (2a - 2))b^{2} + ((2a - 3)\mu_3 + (2a - 3))b + 4\mu_3 + 3 \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^{ 2 } \\ \chi^A\left((((2a - 3)\mu_3 + (3a - 1))b^{2} + (2\mu_3 + (2a + 1))b + ((3a + 2)\mu_3 - 3a + 2))c + ((3a - 2)\mu_3 + (a + 4))b^{2} + ((2a + 3)\mu_3 + (a + 1))b + (2a + 4)\mu_3 - 2a + 1 \right) &= i^{ 0 } \\ \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 + 2)\mu_3 + (2a + 3))b^{2} + (a + 1)\mu_3b + ((2a - 2)\mu_3 - 2))c + (3\mu_3 + (a - 2))b^{2} + (3\mu_3 + (a + 3))b + (2a - 2)\mu_3 - 3a + 1 \right) &= i^{ 2 } \\ \chi^A\left((((2a + 2)\mu_3 + 2)b^{2} + (a\cdot \mu_3 + a)b + ((-2a + 4)\mu_3 - a + 4))c + (3a\cdot \mu_3 + (a - 2))b^{2} + (-\mu_3 - 1)b + (-2a - 2)\mu_3 - 2a + 1 \right) &= i^{ 2 } \\ \chi^A\left(((4\mu_3 + 4)b^{2} + ((a - 2)\mu_3 + (2a - 2))b + ((4a - 2)\mu_3 + (4a - 2)))c + ((3a + 4)\mu_3 + (3a + 4))b^{2} + (\mu_3 + (2a - 3))b - 2\mu_3 - 1 \right) &= i^{ 0 } \\ \chi^A\left((((2a + 2)\mu_3 + (2a + 4))b^{2} + ((2a - 2)\mu_3 + (2a + 2))b + (4\mu_3 - 2a - 2))c + ((3a + 1)\mu_3 + 2)b^{2} + ((3a + 1)\mu_3 + (3a + 1))b + 4\mu_3 + a + 1 \right) &= i^{ 0 } \\ \chi^A\left(((-2\mu_3 + 4)b^{2} + ((3a + 2)\mu_3 + 4)b + (-3a - 2)\mu_3)c + a\cdot \mu_3b^{2} + (3a - 1)\mu_3b + (-a + 3)\mu_3 + 4a \right) &= i^{ 0 } \\ \chi^A\left(((4\mu_3 + 2)b^{2} + ((3a + 4)\mu_3 + (3a + 4))b + ((4a + 4)\mu_3 + (a - 2)))c + ((2a + 1)\mu_3 + (3a + 2))b^{2} + ((3a + 3)\mu_3 + (3a + 3))b - 2\mu_3 + 3a - 1 \right) &= i^{ 0 } \\ \chi^A\left((-\mu_3 + 3)c + -\mu_3 + 3 \right) &= i^{ 0 } \\ \chi^A\left((((3a + 2)\mu_3 + 2a)\cdot b^{2} + ((a + 4)\mu_3 + (3a - 2))b + (-3a - 2))c + ((2a - 1)\mu_3 + (3a + 2))b^{2} + ((a + 3)\mu_3 + (3a + 3))b + (-2a - 2)\mu_3 + a - 1 \right) &= i^{ 2 } \\ \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} + (-468574117940441383158947398788a + 348294242461322892559358706168 )x^{47} + -359157514816171597429507538400a - 532426204140339267735655469268 x^{46} + (-102673599471576936182834808760a + 601600061144731490137800592176 )x^{45} + (4403365123484185126621048232a + 178400864663671072774692222916 )x^{44} + (-440099170390845470333731059108a + 225558282585276428862815804244 )x^{43} + -346825735693224859040991942036a - 214705270630967008269380620584 x^{42} + 331031699335426707894655097188a - 378075465278643382748732315528 x^{41} + -352924016605039237154902100464a - 112006856429892344129818616584 x^{40} + (395413298811543874008162935752a + 630500569998222563985157403672 )x^{39} + (-481073866844916932741022433764a + 193428738413116501845242553300 )x^{38} + (-574890768437172939811699056520a + 429216238452536765703759123220 )x^{37} + 503396162580291811118715650064a - 305335800800334743878139229520 x^{36} + (589805023627727127124027750232a + 160690939463584339975592567136 )x^{35} + 158494449092124449727681631532a - 28334870001524216825093299736 x^{34} + (-385458666717245796933692412892a + 248259833854041663824414942104 )x^{33} + 503148879590858231559336486072a - 457346246192282122949897877816 x^{32} + 411797639576785795463021535816a - 493982544807551235014337273632 x^{31} + -366172927265557417806539902004a - 302907779174134395542162472388 x^{30} + -270813704022870531945228789820a - 315483467222082803047939622712 x^{29} + -227835923607921552509793765600a - 308990254831520507092601163128 x^{28} + (629477798499745468502488795096a + 152599713922387439188249981224 )x^{27} + (-393002493344910685589887504032a + 395767205075320336930567196276 )x^{26} + (119049520455078297910407710216a + 450817292065377802531615579408 )x^{25} + -285329405260475328732782138422a - 410825874902844095600849019380 x^{24} + -564614256178149859261775499384a - 252370421928924763059603226248 x^{23} + (-210051525709496571230940757844a + 297963788674275060661823579784 )x^{22} + (412331788621312132747530569040a + 530898781880845570746210751104 )x^{21} + -27479818289326364024908923396a - 535134985904216068929025621104 x^{20} + (-131776128440286750583506940972a + 137081798275824247818246647144 )x^{19} + -562677738962004557002180167120a - 242870144606571834383545059528 x^{18} + (584981129199571803476355032616a + 627036673749295349028566537384 )x^{17} + (-423066098039623678711211152256a + 474106343504996899912494708368 )x^{16} + (-354515283012376827085533047120a + 200360984650703134118952109008 )x^{15} + (268510333277717975397518535244a + 244895434441192385025871719200 )x^{14} + -91535911723616669727386076988a - 494954092025085828133714539080 x^{13} + (204567683723298982736905579512a + 108958966556849390621408365884 )x^{12} + (-18425180375184448074095094200a + 551777131927397180754899508000 )x^{11} + -336717277495232382701200695760a - 486275950731224580684214842008 x^{10} + (-50949172975629270454373029440a + 148163779415067292353314466632 )x^{9} + (-297020833376766112032848891360a + 400815103801034905887729198872 )x^{8} + -599162383664666671216272329712a - 633099198562140822134209859200 x^{7} + -603716476678263763536340153972a - 545935402677152458901574058496 x^{6} + (576125278821621822765857529936a + 540224840696054369179774832648 )x^{5} + -294439112249222439607443754160a - 629017079137344996422842677456 x^{4} + 267372330032462841984626945192a - 614858129097803174419490106208 x^{3} + -571443199872482471301982786276a - 269559048613704948389647303928 x^{2} + -483812313932110362858617932360a - 375120052556928083569827018944 x - 236641370124149544825479919044a - 121214369440909933885665453546 \)
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