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ex.24.10.1.33_67_101.a

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 \(x^{2} + ((a + 1)b^{3} )x + ((a + 2)b - 1)b \)

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_3b^{2} + ((2a - 2)\mu_3 + (2a + 4))b + 4a\cdot \mu_3)c + 4\mu_3b^{2} + ((2a + 2)\mu_3 - 2)b - 3a\cdot \mu_3 + a + 1 \right) &= i^{ 0 } \\ \chi^A\left(((3a\cdot \mu_3 + (2a - 2))b^{2} + ((2a + 4)\mu_3 + 4)b - 2a\cdot \mu_3)c + (-2\mu_3 + 2a)\cdot b^{2} + (2\mu_3 + 2)b + (-3a + 4)\mu_3 + a + 1 \right) &= i^{ 2 } \\ \chi^A\left((((2a + 4)\mu_3 + 1)b^{2} + ((2a + 4)\mu_3 + a)b + (4\mu_3 + (4a + 3)))c + ((2a + 4)\mu_3 + (2a + 1))b^{2} + (4\mu_3 + (3a + 4))b + -2a + 3 \right) &= i^{ 0 } \\ \chi^A\left(((2a + 1)b^{2} + 2b)\cdot c + 2a\cdot b^{2} + 2a\cdot b + 1 \right) &= i^{ 0 } \\ \chi^A\left((((2a + 4)\mu_3 + (3a - 2))b^{2} + (2a - 1)b + ((-2a + 4)\mu_3 - a + 2))c + ((2a - 1)\mu_3 + (2a + 1))b^{2} + ((2a - 2)\mu_3 + 3a)\cdot b + 3\mu_3 + 4a - 3 \right) &= i^{ 0 } \\ \chi^A\left(((2a + 2)\mu_3b^{2} + (3a\cdot \mu_3 + (2a + 4))b + 3a\cdot \mu_3)c + ((3a + 2)\mu_3 + 4)b^{2} + (-3\mu_3 + 4)b + (-2a + 1)\mu_3 \right) &= i^{ 0 } \\ \chi^A\left(((2\mu_3 + (2a + 4))b^{2} + ((2a + 2)\mu_3 + (a - 2))b + (2a\cdot \mu_3 - a + 2))c + ((3a + 4)\mu_3 + (3a + 2))b^{2} + ((2a + 1)\mu_3 + (2a - 3))b + (4a + 2)\mu_3 + 1 \right) &= i^{ 0 } \\ \chi^A\left((((a + 4)\mu_3 + (a + 4))b^{2} + (2\mu_3 + 2)b + ((-3a - 2)\mu_3 - 3a - 2))c + (-3\mu_3 - 3)b^{2} + ((a + 3)\mu_3 + (a + 3))b + (a + 1)\mu_3 + a + 1 \right) &= i^{ 0 } \\ \chi^A\left((((2a + 4)\mu_3 + (2a + 4))b^{2} + (2a\cdot \mu_3 + 2a)\cdot b + (4a\cdot \mu_3 + (4a + 4)))c + ((3a - 2)\mu_3 + (3a + 4))b^{2} + ((2a + 3)\mu_3 - 1)b + (-2a + 2)\mu_3 + 2a + 3 \right) &= i^{ 2 } \\ \chi^A\left(((2a\cdot \mu_3 - 2)b^{2} + (3a\cdot \mu_3 + 2a)\cdot b + ((2a + 4)\mu_3 + (2a + 4)))c + ((a + 3)\mu_3 + (2a + 2))b^{2} + ((a - 1)\mu_3 + (3a - 1))b - 2a\cdot \mu_3 + a + 1 \right) &= i^{ 2 } \\ \chi^A\left(((3a - 2)b^{2} + b + (2a + 2))c + (2a + 2)b^{2} + (3a - 2)b + 1 \right) &= i^{ 0 } \\ \chi^A\left((((3a + 2)\mu_3 + (3a - 2))b^{2} + ((a + 2)\mu_3 + (3a - 2))b + ((-a - 2)\mu_3 - a + 2))c + ((2a - 3)\mu_3 + (2a + 1))b^{2} + ((3a + 3)\mu_3 + (3a - 1))b + (-3a + 1)\mu_3 + a + 1 \right) &= i^{ 0 } \\ \chi^A\left(((2a + 1)\mu_3b^{2} + 2\mu_3b)\cdot c + 2a\cdot \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} + 118316769627035975998551516572a - 238598229279484462010323883136 x^{47} + (419208574292593586227565020572a + 432640597491989782687473869388 )x^{46} + (479475600487656107319008056456a + 532100912216282346155455142552 )x^{45} + 362444535699513918386980066636a - 449295969898359917880077657964 x^{44} + (115903144781441241689343571116a + 536479974955304199091891641412 )x^{43} + -463341141559814412460188873412a - 590624143738355205535848299608 x^{42} + -14390670852374763700971750012a - 301979237883415174143538516424 x^{41} + (-2373005153485273420773292572a + 40297703275365444831010067856 )x^{40} + (-500371306327284813097355126984a + 234985451140266796403034663976 )x^{39} + -100661515683771797904881895572a - 11420654258379701669515530916 x^{38} + 183747466621733499108798901224a - 166451821714150203299941235876 x^{37} + 549422735430040830333630433544a - 438093407981906391276367774736 x^{36} + (-49560967754508686987803481264a + 320866278786696077317477440216 )x^{35} + (-473971814503114409120543612632a + 418357986581365653969967373384 )x^{34} + -468738982628374614704622518420a - 567094980655565432744887385688 x^{33} + (352072536141750427563785556268a + 416635850727925893372399488864 )x^{32} + (-362070419027572656880815987976a + 239126788559281731839856719632 )x^{31} + -265819244934291816293261500584a - 361741613738146148533119409668 x^{30} + (-454129454512859690104301932728a + 250651153717772127746076241312 )x^{29} + 141414764714337893310390225960a - 450346532031474847914551751704 x^{28} + (-139226243313388173441032352072a + 136529097622121659845020857672 )x^{27} + 339452701908085208454549461904a - 18294010622175769499303106876 x^{26} + -185384929846349155382012509752a - 168819403628545531924233432600 x^{25} + -56189307938916239924067634450a - 564932117251668392281333863140 x^{24} + -42569884377337158161520544216a - 283247226862753600141167746744 x^{23} + (-119534556672547684756869419380a + 254158464859432037673705522384 )x^{22} + -110782711609268693641061853608a - 137711217257747435609140414512 x^{21} + (451125797812364499642911810844a + 141077641368903402972202031288 )x^{20} + 204549133939040403915179108060a - 151266205549316535256808118088 x^{19} + (-488501197436758771518091628728a + 402082240413150990740519352280 )x^{18} + (-562011649494202333230436407616a + 382865205758659625186273675256 )x^{17} + (627029978582294376852692754176a + 490485607829834738941567868792 )x^{16} + -452162941904965247187438979648a - 256269557014855744297243071648 x^{15} + (-608896982604408654209124493460a + 409316830628516165281684414400 )x^{14} + 262919123437274272348678789460a - 221554256587913496476545081432 x^{13} + 384041385798508172152746603056a - 602977638537127577832751182940 x^{12} + (328830528743563418301795569032a + 221708005917537467987467098432 )x^{11} + (429527947251251432879315837064a + 342528734931738958427886264032 )x^{10} + -570895081533933225115410101896a - 32921787753861210622657087336 x^{9} + 429978973727944296405077294528a - 553991587519457250862815057072 x^{8} + (627152596195413368446224773792a + 303225045865469664562602406160 )x^{7} + (-21017212148526884491567964436a + 633595508908061847981981913512 )x^{6} + -398341384149060693399448272352a - 210677191615449020978676657024 x^{5} + -208525485822153095947957147504a - 570515940781845095704645062336 x^{4} + -167663310042771889842565259800a - 424030762843273909767191678176 x^{3} + -418228835029625149173010891676a - 299038187041854604864492466104 x^{2} + (-520299929551050679690731504224a + 616423120359498021628900896448 )x - 182975962339052632619594190974a + 97179721347928341939422666706 \)
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