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ex.24.10.1.31_63_95.a

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

Inertia Polynomial

The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + (487301373115530902857108970204a + 137716051067326585018195009080 )x^{47} + (404668249342939117265935342692a + 179502606459815994220708301944 )x^{46} + (-379840444991790164327716027432a + 429966885057528290794220789968 )x^{45} + 335424805167302116432192655632a - 537966883304239603958859748344 x^{44} + -14445846474244736900939923248a - 166331754991636965696368412516 x^{43} + 328222704743184777565273229400a - 117909754436604014427186202872 x^{42} + (-545585066163383694074631032540a + 68774223766775185448769999656 )x^{41} + 556635256569605016413966207056a - 156371294904784483349837112616 x^{40} + 233223354339027008772165318104a - 39814145849034880586790959296 x^{39} + 401088544925130697657702886280a - 133567837785826869624357638524 x^{38} + -474250044859524022120844410772a - 431557938022603070077810976064 x^{37} + (-515428949349997622854810969844a + 507538271382883897236551749376 )x^{36} + -85245366252501168037609447304a - 299252754444689170797990611008 x^{35} + (236363436993415567325055023456a + 380026409784911015208894012512 )x^{34} + 342618387585038298809140718376a - 115607453997962335200102544520 x^{33} + 141370560129184956390430287320a - 241538078054335688897111794032 x^{32} + (-399532286648957048964945538596a + 149999045491745658426476272560 )x^{31} + (-414538087876479039017147387704a + 511050575102334983630637503164 )x^{30} + 142861065734694942156833062572a - 195355262480826968580168897952 x^{29} + (40565834294878910349623434264a + 522646477414423334320123197400 )x^{28} + (551666196766475420387756017728a + 433451723580481373831415983776 )x^{27} + -120010627034574453055649881848a - 17548143917978301699252315904 x^{26} + (277570966206140506747749537248a + 539377427572591703087799445328 )x^{25} + 28047665899842531348800537982a - 77663400072147653152808812040 x^{24} + (545562139178061376786252882096a + 525605266839480246591904949112 )x^{23} + -321775525531222772182308539672a - 254240970408813193008198656264 x^{22} + (222258760191374945564431600768a + 497271832273604432570494270128 )x^{21} + (30387404767617505692298844896a + 617775782625096409859023581128 )x^{20} + (527318418515119878218628197788a + 562788992744851202510088834384 )x^{19} + (328500720682610736260887760848a + 323815119751795446057866505696 )x^{18} + -18515917242562019774888456504a - 236233757013875617385165113992 x^{17} + (-81892263430421239493751389400a + 287020079664756071835348981424 )x^{16} + (406511799669897795222556021968a + 167106771630160863300546081568 )x^{15} + (76794316149411651805137928940a + 260504131693895749873477092720 )x^{14} + (-304827404322174186189658261560a + 407028515858938518435516834664 )x^{13} + (542338373649417812359512468088a + 463109656262836543173944128956 )x^{12} + 110206592634291058896568876808a - 97879326436975979900823031280 x^{11} + (590467950539614079293299924160a + 415129291464181097176595280704 )x^{10} + (461464090015868791231483444808a + 486764090174823014366729756976 )x^{9} + (-206097605759227867163101495488a + 496801140886725954835963885440 )x^{8} + -588578269871919299660122025648a - 133779586099637385343982935048 x^{7} + 474098320523809093879613492332a - 51174831307024446061552651648 x^{6} + (-268631769701702852092887634960a + 314199335926958485662339636632 )x^{5} + 617193109346538932962709744024a - 39642330207035659317926224896 x^{4} + -166786084353512608037763741056a - 131392131784208682747114452640 x^{3} + (-390669585816761398451084252976a + 83055441527688329971728169888 )x^{2} + 353329157007772418004657920264a - 329332663353849147531703801056 x + 531607714534650517594774608800a - 95749409060795956625541069354 \)
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