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ex.24.10.1.131_259_387.d

Base Field
\(F = \) 2.1.2.2a1.1 \( = \mathbb{Q}_{ 2 }(a) = \mathbb{Q}_{ 2 }[x] / (x^{2} + 2 x + 2 )\) 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 \(x^{2} + ((a + 1)b^{4} )x + b \)

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

Inertia Polynomial

The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + (-474211381130228606402700099640a + 362793698867465405522521801784 )x^{47} + (489818290337977268331948903416a + 162525129522511156204669932500 )x^{46} + -554076345619608143713585337604a - 419508258961695076456854426720 x^{45} + -540731354672022378961520716532a - 589928994842042913765109534688 x^{44} + -32519042570962177421856220056a - 413491814040525446006013962416 x^{43} + (389501386746573855368187531096a + 127880066412703138711205724540 )x^{42} + -205634507229046289813628920952a - 16013607056295931171722302200 x^{41} + 120036311093165849998728822540a - 294487531464534331792234952564 x^{40} + -266248436250207930774254372968a - 248638482814040643612215043904 x^{39} + (125065166245862411253920300240a + 392321706007554595289942717424 )x^{38} + (177328951138019477637749863508a + 308067023905985584117762959752 )x^{37} + (-99624941691502922701648138584a + 212990085744292974810571329388 )x^{36} + (-414230908489600521239108162320a + 106489967333704221114958441408 )x^{35} + (-238170596905754650036650374988a + 446626428633918934787505174920 )x^{34} + 467604240106885708555029655720a - 113733859228374671573604370800 x^{33} + (53629047678311692354892629356a + 494015473595723201225298729376 )x^{32} + (526891273755153791462683521888a + 485417004203264151708860847008 )x^{31} + (-538324969547888404397538701768a + 500166285047689839755426076752 )x^{30} + -410549216398739485100763833872a - 201755139349636651129800995552 x^{29} + 540078384245311033186331574540a - 487913586929411855871152228752 x^{28} + (175342036644923560104226416768a + 204379764956512542438677977968 )x^{27} + (-405417607332271835376629852848a + 522974243247696687739762971144 )x^{26} + (-447943997792582796590509186896a + 429095306728738698447405844912 )x^{25} + -232148348547604209591358892954a - 75662953086830018244994521516 x^{24} + 530691485997680499994559709208a - 433598149383235797893552562976 x^{23} + -208050580992037201156078608604a - 172140568452633945698338184200 x^{22} + (519574626222972853846720587376a + 183975538685299748420974328408 )x^{21} + -543050397062607187364892844944a - 357401421607546365152831791352 x^{20} + (-321616125169262320974449972288a + 50320339412024628072510283408 )x^{19} + 406591293008996483276175308732a - 2699336889201427729805719912 x^{18} + 268926825214345658048777227000a - 31491461673339959821367787808 x^{17} + 58684610915529105714159940292a - 173327082721510661310136547536 x^{16} + -106297736806314782196998784016a - 570279411793009298937361262688 x^{15} + -570628028518302522675759731760a - 167958002200559066617086893216 x^{14} + (-94252521104002034785777687576a + 137123053142309714505885791960 )x^{13} + (-176455266870063936114374857552a + 447317399173616262080291561720 )x^{12} + (25293570809172299334493043024a + 472663657617750632118574628960 )x^{11} + 173462767417799868803171391712a - 382610324174478873734544647320 x^{10} + -25897242829209930682619711776a - 357489111394461756214483693520 x^{9} + (-264050671385583181949902433616a + 500737327759501783077564389256 )x^{8} + (-16045650635140370263371064048a + 279609450358957682318465426096 )x^{7} + (397669804882756998689772406976a + 594412214370749323469633175840 )x^{6} + (522828102562466771523829361552a + 136413022761647736840934674304 )x^{5} + -365067735290982802687029078048a - 394297765785233766894819594824 x^{4} + -325301446878358024940297059088a - 407322159597760142748390982336 x^{3} + (-336560144842064715012319993256a + 417851289647771901008954934048 )x^{2} + -129690624392011449956066180320a - 429569115345241704877379124512 x - 114522114776107654808317106006a + 76785977712693905711333178394 \)
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