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ex.24.10.1.31_63_95.d

Base Field
\(F = \) 2.1.2.3a1.4 \( = \mathbb{Q}_{ 2 }(a) = \mathbb{Q}_{ 2 }[x] / (x^{2} + 3\cdot 2 )\) 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 \(x^{2} - 211275100038038233582783867563b^{3} x + (-211275100038038233582783867563\mu_3 - 211275100038038233582783867563)b \)

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

Inertia Polynomial

The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + -23290113418896581130284151300a - 379227890249476545421003768528 x^{47} + (327359080442854713394847898916a + 229760200819687523943282307920 )x^{46} + (-556917232100972485092419753640a + 552799154678718211215038750424 )x^{45} + -296178597361234792742076418948a - 50753572676667572836977653632 x^{44} + (107601609569681981871100301072a + 424073206661857754487695054700 )x^{43} + 302133858070150155205978654196a - 538893135501056055971665174648 x^{42} + -335417237309904359442552277864a - 529182852724793002274204451000 x^{41} + 112585526701003236745293536920a - 608759337182307366073492182456 x^{40} + 483863513561012868140866041104a - 418509126275943494458536671760 x^{39} + (153326866269825528180961384136a + 384259668654392191127603152356 )x^{38} + (595416630387034428367624463056a + 363976130951564766785798367728 )x^{37} + (206211970025293918699360301900a + 474050839002947040867751060992 )x^{36} + (-622486963351136313071742134128a + 532492198617227396987802204584 )x^{35} + -302107600499502322996926641836a - 480900972002905892256504534576 x^{34} + (-426698979495724517166467455288a + 343402839553079987257719631688 )x^{33} + -111606782928737118364646833068a - 556741314851313144238902793816 x^{32} + -407536455900705201479872339068a - 204482538359877007865571885520 x^{31} + (-355100555177507549276395839416a + 372657886790247197218363572676 )x^{30} + (-142659620466446202046006403416a + 481501940730083875067216280960 )x^{29} + -593717009197670594584678419432a - 547329866174533833317108871664 x^{28} + (-531721128039432394284140278416a + 633339927925270872971848611520 )x^{27} + (525707016651143094316608722352a + 237027093714295705949759277048 )x^{26} + -411065220575947254913893008600a - 627060641721397195985294041216 x^{25} + (-95670079218623714413505093746a + 350925999084609315285618297512 )x^{24} + -560104509996640677705278513520a - 270575233560376838399209412936 x^{23} + 114218711339006194833010632528a - 154328908863479832737446591496 x^{22} + (-369650599113349752451105823016a + 513206749615270041579227966032 )x^{21} + 363513746711505301550440850376a - 242181171733832070981703940608 x^{20} + (377341936699386082168256234468a + 49474952954988350583803985200 )x^{19} + (77930463026083790420149442032a + 250706678776791091717772888584 )x^{18} + (552831312026282700253794697608a + 563411338472345250714522204672 )x^{17} + 102183688681520487762367300208a - 535510167569863240160699131360 x^{16} + -372528463381601897237273945488a - 487294932389832451503865310416 x^{15} + -508806980799834116628990043580a - 130442486879701824469078839360 x^{14} + (-530571632394562440022896363496a + 553140907601902348626585672672 )x^{13} + 557790811240761651058419576528a - 516370969684004451489267890388 x^{12} + 170353154251062122447674940344a - 381202356894155165295365780032 x^{11} + -137635670047120871292060711312a - 234496446116601976946590339960 x^{10} + (473486723531732836462518165176a + 321783247710344045901550899856 )x^{9} + (390912710104775976191309696168a + 274049332769722774929234458408 )x^{8} + (325123660384875144461332070608a + 3710743908170251762596704280 )x^{7} + -187228604118184853857728347100a - 77650509130474391486815564224 x^{6} + (521139998990590791796734548688a + 8162290967062954634433258192 )x^{5} + -361654364895204223180938355504a - 214940187389308931632679011008 x^{4} + -160387676065656678966334562816a - 540882203745195006843734597728 x^{3} + (533197201283819477256248893800a + 182218747833756933867737068128 )x^{2} + (211942294087703614432917581368a + 120839140371603719224540660112 )x + 459230378104210013945222840640a - 150085411289205180971993720534 \)
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