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ex.24.10.1.131_259_387.b

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 \(x^{2} + 90546471444873528678335943241b^{4} x + 3b \)

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

Inertia Polynomial

The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + (-403012332264745202745056448560a + 182443817911977537905478881992 )x^{47} + (-415634886699058949868563432096a + 587758726269696306017445218972 )x^{46} + (-170755570797329825242887331980a + 506477324776649790800487373552 )x^{45} + (200001105026070988016896848696a + 12719974358112565965158961680 )x^{44} + (362282748808629255260402082264a + 508364155289301749187854960224 )x^{43} + (-596745056654162004213270854484a + 251776763836896157382610502668 )x^{42} + 614026029564918138214057320616a - 2709887795280170158830788536 x^{41} + 75057229022656665430443699704a - 613194938763826774908569963004 x^{40} + (-338724492060087179437484564872a + 384835178838605127548363836192 )x^{39} + 276539740642246917917437391608a - 59563548998432646704280517176 x^{38} + (-140867283438461551045519370468a + 559971401389661104688903035848 )x^{37} + 276422867524563343226215062228a - 573520195507501892253050484020 x^{36} + -604170504844945695549123305936a - 488768916412354921364366245792 x^{35} + -121926509544746189787634977628a - 613831608347252048086014257568 x^{34} + -212662626208860827346801659728a - 525832143251032365220322247928 x^{33} + (370405784512777925715029599312a + 162585616674785675269533135680 )x^{32} + (248637542314312692518570726400a + 196495418969237866461510970112 )x^{31} + 625777953117428142503595401424a - 173028285922657509789262201520 x^{30} + -311024959078997744797952488448a - 124095975606144616022730085472 x^{29} + (519306043086943518394059504920a + 170030847498745234098356082232 )x^{28} + 450111788274111281659081051440a - 182452907772930517821930519840 x^{27} + (-460100878575445503454944544928a + 51785799346850677858992198472 )x^{26} + (-241987970370178487507596340328a + 435070447861443910810104331920 )x^{25} + 218414280024909162061084616506a - 203450748240257873327887313268 x^{24} + (152263222854889539309185335960a + 555697442592233901853368839792 )x^{23} + (475478313578123277029813990700a + 408919358895178922588890407336 )x^{22} + (310519672178157550275628065368a + 227771299095036944683594414440 )x^{21} + -52346273387872852116072368304a - 584179981764744050139255462112 x^{20} + 153207664240440828202744889808a - 559544113773601851946664723312 x^{19} + -275104169765749780768788827300a - 560250756529484847153261976736 x^{18} + (362671137196497847533064635128a + 25348052459079298113760916992 )x^{17} + 282545345725185303154364476020a - 262796227569407305725429368664 x^{16} + (-612846681137977339796533877376a + 585308473831089040421493731936 )x^{15} + 412783325568675377499044361096a - 20381518978581135698116711808 x^{14} + 121813579164766110095750706608a - 251364199373219944068663577080 x^{13} + (-205856272673341103622546721272a + 308745885654454274711954985376 )x^{12} + (216552154852129638944602209872a + 53210124412439902604011388768 )x^{11} + (-69184313399608416651769611680a + 194220702913348647558016380472 )x^{10} + (138540685126948297932004367912a + 521093951618807362824561341616 )x^{9} + -331089841532859086365060763856a - 117619373813435862212120302496 x^{8} + (-289562622627568566227535343776a + 44453897962955980936717226352 )x^{7} + -590791852665867165789445845184a - 443153411392978368755289404864 x^{6} + (376875435709233260508097453024a + 208448136237044780942936805344 )x^{5} + (-213356949674863917312754866600a + 73331068492773533913167806784 )x^{4} + (577979003391856573711288642416a + 369997974775569584631370517632 )x^{3} + 266193848656678955185226442424a - 397107824213137431648608443632 x^{2} + (77166118076488724995731320448a + 513501204741946708220080182256 )x - 436701894994972598216045615732a + 478098855919981759838934209390 \)
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