← Back to 2.1.2.3a1.2

ex.24.10.1.31_63_95.d

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 \(x^{2} + 253530120045645880299340641075b^{3} x + (253530120045645880299340641075\mu_3 + 253530120045645880299340641075)b \)

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 \((\mathcal{O}_K/\mathfrak p^{ 20 })^\times/U_{\mathfrak{p}^{ 20 } }\) :
\(\begin{array}{l} \chi^A\left(\mu_3c + 1 \right) &= i^{ 1 } \\ \chi^A\left(((4\mu_3 + 4)b^{2} - 2a\cdot \mu_3)c + 4\mu_3 + 1 \right) &= i^{ 2 } \\ \chi^A\left(((2a + 4)\mu_3 + (2a + 2))b\cdot c + 4\mu_3b^{2} + (4\mu_3 + (2a + 4))b + (-3a + 2)\mu_3 - a - 3 \right) &= i^{ 0 } \\ \chi^A\left((3b - 2\mu_3)c + a\cdot b + 1 \right) &= i^{ 2 } \\ \chi^A\left((\mu_3b^{2} + (2\mu_3 + 2)b + 4)c + 2a\cdot \mu_3b^{2} + (2a\cdot \mu_3 + 2a)\cdot b + 1 \right) &= i^{ 0 } \\ \chi^A\left(((-2\mu_3 + (2a - 2))b^{2} + (\mu_3 + (2a - 1))b + ((-3a + 4)\mu_3 - 2a - 1))c + ((a - 2)\mu_3 + (a + 2))b^{2} + ((2a - 1)\mu_3 + (3a - 3))b + (-2a - 2)\mu_3 + a - 3 \right) &= i^{ 2 } \\ \chi^A\left(((2a + 4)\mu_3b^{2} + 2a\cdot b + (4a\cdot \mu_3 + 4a))\cdot c + ((2a + 2)\mu_3 + 4)b^{2} + (-\mu_3 + 4)b + (-2a + 2)\mu_3 + 2a + 3 \right) &= i^{ 2 } \\ \chi^A\left((((3a + 4)\mu_3 + (3a + 2))b^{2} + (3a\cdot \mu_3 + (a - 2))b + ((2a - 2)\mu_3 - a + 2))c + 4b^{2} + (2a - 3)\mu_3b + (-2a - 2)\mu_3 + 2a - 3 \right) &= i^{ 2 } \\ \chi^A\left((-2b^{2} - 2\mu_3b - 2a\cdot \mu_3 - 2a)\cdot c + 3b^{2} + (2a + 3)\mu_3b + (-2a - 3)\mu_3 - 2a - 3 \right) &= i^{ 0 } \\ \chi^A\left((((2a + 4)\mu_3 + 2a)\cdot b^{2} + (3a + 2)b + (-3a + 2)\mu_3)c + (3a - 3)b + (-a + 3)\mu_3 \right) &= i^{ 0 } \\ \chi^A\left((((3a + 4)\mu_3 + (a + 2))b^{2} + (2\mu_3 + (3a - 2))b + 2\mu_3)c + (2\mu_3 - 2)b^{2} - \mu_3b + 2\mu_3 + 2a - 1 \right) &= i^{ 2 } \\ \chi^A\left(((-3\mu_3 - 3)b - 2)c + (3a\cdot \mu_3 + 3a)\cdot b + 1 \right) &= i^{ 0 } \\ \chi^A\left(((2a\cdot \mu_3 + 2a)\cdot b^{2} - 2\mu_3)c + (4\mu_3 + 4)b^{2} + 4a\cdot \mu_3 + 1 \right) &= i^{ 0 } \\ \chi^A\left(((-\mu_3 - 1)b^{2} - 2b + 4\mu_3)c + (2a\cdot \mu_3 + 2a)\cdot b^{2} + 2a\cdot b + 1 \right) &= i^{ 1 } \end{array} \)

Inertia Polynomial

The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + (204877173336531253610404736268a + 534322942957277372268937771760 )x^{47} + (49139298506475596999859425812a + 406919581313431357437212516376 )x^{46} + 139851683396837022495687028000a - 313445292045759431292411891256 x^{45} + 530282523890078537387319163316a - 621063167262729527020214644232 x^{44} + -183841815821507482020663784040a - 497518207286101112008115828404 x^{43} + (182015880660378268983788387536a + 553287158713555070968936051416 )x^{42} + (49605622546748851199823168360a + 55651125539860175027972828952 )x^{41} + (163323134537331161254746288488a + 139848440144835640487868485728 )x^{40} + 520377338650880931821518505304a - 425077068328909827379968931408 x^{39} + (-543818728778918833997773200752a + 481064345292585731803941199524 )x^{38} + -150297565634095339312712577084a - 478655162924130495543669702592 x^{37} + (376778421348645764131973121392a + 106534055711453411614954098552 )x^{36} + (463241799454099056371765008904a + 440992059249642731817292820768 )x^{35} + -120936482012644800700820384448a - 275215129905903432710768554352 x^{34} + -6544379841991903277160951368a - 233240126262246518149577303176 x^{33} + -64350275574903293437005060776a - 417175967985268533285199753592 x^{32} + (507910703091486050606628248336a + 358573836666626158690134678304 )x^{31} + (-561932533324514709276164028288a + 620474088718895142155695372044 )x^{30} + (110651744036950892881663285852a + 216619156060627825818494830832 )x^{29} + (-168349465146348641452761591632a + 90105987179941645261456947512 )x^{28} + (-5209982512576215942525550496a + 601431977145164462109850647376 )x^{27} + 525816523429865023360234237640a - 93671271644864877288122339728 x^{26} + -156120863967676372904535474240a - 61433374355262541698795023088 x^{25} + (139995528266992923851432851422a + 130749487176596884405241589592 )x^{24} + 416793366327820761187472929536a - 488205550707985002399898797864 x^{23} + (435151123787099332179718424408a + 8738711570665260381222600 )x^{22} + 484172898004663010856878647992a - 155209017264967895300059509888 x^{21} + 473527651167175534909840866416a - 117363676807535298477223733360 x^{20} + (316980274686520786063421491676a + 476028244645888350756159506176 )x^{19} + -415285066427762402716645273816a - 247477855935524992601195276416 x^{18} + (-48801971121316245285193648408a + 406284136821078107235394510688 )x^{17} + (42186588810216314355904041008a + 184397624884397101824861002496 )x^{16} + (286236218475703883112695356288a + 73978590631403749886084909600 )x^{15} + (-4153174405622930689450313396a + 375869938305825053660113115584 )x^{14} + 447640642801156533350595302352a - 147140554751027062768587782184 x^{13} + -178604948799974681706271301152a - 631504529694706153136446188732 x^{12} + -86858989563868507319713101752a - 559095681438240711964263926640 x^{11} + (-345873843448968854547806067936a + 324571538795209354086671243904 )x^{10} + -19086826080899015537764360856a - 177090585065237921379509406096 x^{9} + 589296524988036753001081935800a - 257134165790662268634217223072 x^{8} + (-74872723214837840176044105840a + 13268769035199658797082378320 )x^{7} + (272600832601670982360088621660a + 345575842764896483169382120496 )x^{6} + 13797592474235899238632757920a - 223697381933794600286676841640 x^{5} + (549871272311725947413201690936a + 50298543877248473335093863024 )x^{4} + 507080035107521632955009425696a - 325408082447623349843971877376 x^{3} + (-66515374441103123117546865104a + 352018668566661258334415469312 )x^{2} + -110229785119595342214269857272a - 614619534014192579475464884096 x - 483058888989716362551537621832a + 370952558793176782351764354374 \)
← Back to 2.1.2.3a1.2 Summary