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ex.24.10.1.33_67_101.a

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^{ 0 } \\ \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^{ 2 } \\ \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} + -238067837838019993960529701696a - 593480996043203622202417394496 x^{47} + (-464182086227557635680778387852a + 225933152209702336093799613660 )x^{46} + 571828675049616498265388254940a - 78227517664766674776794858672 x^{45} + (-418727877300679710989579897140a + 444596781764058349007926868900 )x^{44} + 143258709118389865564140771804a - 135136834986884519208321777500 x^{43} + 19205113458799122548121456900a - 336800993655086384721927270352 x^{42} + 384282422727871010259545461744a - 573261755886333144089689021336 x^{41} + 38033878727221489409490309740a - 395563431165590598296525637000 x^{40} + (-118076452310141436909746934256a + 422590801196259830452045023840 )x^{39} + 330823162414462859956183049892a - 528410516394012306089608893508 x^{38} + (-483775802112909801047945217632a + 536653457392051914908044099828 )x^{37} + 509841971852632564656210597244a - 301292039730275518233415040008 x^{36} + (335275648443712666065338833496a + 118587900638347468961962890008 )x^{35} + 256886465431371540269299789868a - 468857776857026645577749818800 x^{34} + 122822078056271982205601624524a - 62367811023995435194736467336 x^{33} + -420269963781050905441625461012a - 355635348481502468630914756920 x^{32} + -178901185033671277271055232916a - 533908040495936344022598137184 x^{31} + (39943788594238858606204909372a + 521701804679357144983574036124 )x^{30} + (347202067464688626416511235092a + 149837987373277049497818412928 )x^{29} + (459178896464603502529735168992a + 308114077358027173194902748200 )x^{28} + 507124854251884368213928379624a - 364645081823548623558806065752 x^{27} + (619254653518191564695036861656a + 574082180157111344296001159716 )x^{26} + (448545297310286917133335950372a + 239953930318527302902874921688 )x^{25} + -59955333073865067593659988030a - 42082392775929532226927315500 x^{24} + -616193600069909729077013608688a - 498649058132678142733682936848 x^{23} + (21839391898550199833060816764a + 56916232412738435841891998064 )x^{22} + (258720990104839610156529502408a + 146001348847796565238192960872 )x^{21} + -619959071046775997852086633292a - 504271626921154902503988412376 x^{20} + (-344971459951459613120095297404a + 4320923320585731863467734520 )x^{19} + (-172071343985891251702189618656a + 249810855720468877388997666136 )x^{18} + (548272655466764203312218429056a + 488656384390305608470403724928 )x^{17} + (277028589176717426006197677704a + 230515720143702533612539244504 )x^{16} + (-148531795414028207655981707368a + 503128067069741089739858071120 )x^{15} + -442874308147612394165910356588a - 586410124976134809280300318528 x^{14} + 86267777557073666733155173756a - 370161990805458266674464382776 x^{13} + 170781997287366850778185503544a - 115583790155584833993756491484 x^{12} + (-399099320793207272890561988160a + 52344695128142766330279720256 )x^{11} + 313994649249532079127383092888a - 122295288226256933101358439256 x^{10} + (292512883647305391735386191728a + 83581954940307687103906981400 )x^{9} + (495919490797583655386486715304a + 292759601253262174498047122192 )x^{8} + (559540372154995398312359317312a + 127269903500226982553798185560 )x^{7} + 302546398975027925844807379484a - 501850916973851266564918425968 x^{6} + (-410982278531624741600605254040a + 370917751220155343072170632920 )x^{5} + (153467047052531390647876643536a + 351076509687576634421328975168 )x^{4} + (398910434690861794648660625400a + 114645854442887337454681115504 )x^{3} + (-320081457835193847358352558884a + 89987492944111926088982627256 )x^{2} + -442066082171685345483572828968a - 325257102673111894215742122856 x - 583245748097256552118566123028a + 504473228926448365931345657606 \)
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