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ex.24.10.1.127_255_383.c

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 \(x^{2} + 253530120045645880299340641075b^{4} x + b \)

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

Inertia Polynomial

The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + 193493051884295649913944560368a - 166048338844745378223307361960 x^{47} + (193971843191827717286214654196a + 176708995442476878649908781904 )x^{46} + 324607586525552336509598799564a - 9045957090438060497708856112 x^{45} + (-535109200618765100903894429504a + 616259093932052985852844593100 )x^{44} + -255920474615378089727454069856a - 443044723702544432389741503840 x^{43} + -127127218329058403763394013460a - 529701571772599123875993918500 x^{42} + -481800554517542407182002215496a - 212212606620177251713131288400 x^{41} + -244397580647410138721422172504a - 249273269306858023216556496028 x^{40} + (418040105123768974675509046520a + 9313929720616979492392804064 )x^{39} + 617277759612967635705991813500a - 384664516599896898238916857328 x^{38} + 466480304733846005157839852444a - 202691999265043877531454940576 x^{37} + 395463932744687505092532638076a - 323344303520401076091478764860 x^{36} + (17957114713165617077618679928a + 341153660123007514365203422368 )x^{35} + (129803057871422116391320281700a + 130787698126522469679047605544 )x^{34} + (-159816674992847689872599645160a + 261184121140790600618079164640 )x^{33} + (-493446272021123207027195694492a + 69436185411503481615790046288 )x^{32} + (230332697689211529227107618272a + 273240443976800419723410741280 )x^{31} + (-581543316534229714591122826880a + 48605395288387550119444756848 )x^{30} + (512323154159777516001442022312a + 308695840242857952972274620008 )x^{29} + 510859973443756732707906129028a - 615992276212387984229763135776 x^{28} + 140595129649374907322673521600a - 566624340991943630638391136816 x^{27} + (-45696820531090194138577746796a + 338657620803361721584301403664 )x^{26} + -138281361371341301890762227160a - 44935808798482582660251165808 x^{25} + -89852135589584902968886287338a - 81424214938020635751062321768 x^{24} + 464326482036194321176549623336a - 361222961292386510254274684608 x^{23} + 424404788662582185717152397488a - 247342926388782195102220779400 x^{22} + (-19825466774976871845589775440a + 232720906382473862384260021912 )x^{21} + (-612272901275933753682991504956a + 75836968244514822248863996080 )x^{20} + (508255870678942193071132371744a + 91156412978916654653513030576 )x^{19} + (249785691871699382172480889628a + 69953861123039884692956246344 )x^{18} + -413235104335734918899152840528a - 62629313625732318574415756256 x^{17} + -95266226652299519308676658204a - 472778446017838801136644185152 x^{16} + 520699070132805686270084438240a - 502351208386275914652720306624 x^{15} + (-57328344077777203676335864888a + 2926992424304662594524203560 )x^{14} + 153276969592147391304598954272a - 388536045086344621516027541000 x^{13} + -6758128373267222406698468984a - 385245649080367011230758581328 x^{12} + -510485700351086745732763332912a - 52275474343381026379592251440 x^{11} + (-151230001242379617568288318240a + 550311710183088543575033340440 )x^{10} + (274781459144533864456344751040a + 135262364834388290439525400592 )x^{9} + 11919549359140631077464647872a - 182808530862472440329210718312 x^{8} + (213720050801637667500560031840a + 103127318988517445982307220304 )x^{7} + 245620841944700290674448364320a - 204504730406018169431971079808 x^{6} + -282456614213257364975855081240a - 124538897775666705885482547632 x^{5} + (596193274630172662278543231376a + 136854841351298374253364626760 )x^{4} + (-245728866921889143961861778544a + 593361963137904172778932055456 )x^{3} + (607707930904989181304118521984a + 191528760823165673087395393944 )x^{2} + 206219737611704397040863536144a - 370646050956659629122753109360 x - 581455633316334206115292083800a - 112837116202762734339502202466 \)
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