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ex.24.10.1.31_63_95.c

Base Field
\(F = \) 2.1.2.3a1.1 \( = \mathbb{Q}_{ 2 }(a) = \mathbb{Q}_{ 2 }[x] / (x^{2} + 2 )\) 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 \(x^{2} - b^{3} x + (-\mu_3 - 1)b \)

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

Inertia Polynomial

The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + -526685287819304360435490361860a - 54771264462857055821212159096 x^{47} + (567621633468508302978466949484a + 158255834351579651430781310888 )x^{46} + (85725540280033004352038595416a + 559609553277908915000362730496 )x^{45} + -582180476507547024200360971432a - 311918671554253518421724199368 x^{44} + (-222801965522574406933635396560a + 620217963979204096792808841660 )x^{43} + (-519249767133512469015827175680a + 319269590650157157343136457512 )x^{42} + (368549699187286039677220900924a + 24961076399529834786005647544 )x^{41} + (-423178896262631238692810823888a + 338945403052199422661992395528 )x^{40} + 225879188041809941432993813768a - 258525408791810247611313439264 x^{39} + (245943114684402424198176058784a + 437921619341606942739488065412 )x^{38} + (472177147482155676020039945580a + 254962357685416459900968421176 )x^{37} + 181847632982156687520790963436a - 229131031616379336539435916512 x^{36} + (-202215766912168738807403799976a + 298888262612509685009449892368 )x^{35} + (-212858930638559289730128801792a + 554088268136368965659307394816 )x^{34} + (-423896821718785714459794197720a + 101674122925436943927496307576 )x^{33} + (463430306289581564660046325200a + 192619670365341027524486175344 )x^{32} + (-492402838027829575938282021732a + 161232286730326531593975182576 )x^{31} + (-259598471945973828399056332552a + 546441029586633887058159748204 )x^{30} + (252618631918909404992791544692a + 333350409063218210016271146496 )x^{29} + (-581647929159068476478248692600a + 517185421431117906102592265592 )x^{28} + (283401887958568396064623958656a + 234689039301925142679336454016 )x^{27} + (-586372845660499770022552011192a + 152501242329009223813457967152 )x^{26} + (-86253468970977866604531841208a + 622867347429141078376971924832 )x^{25} + -57048481931559186866436448650a - 238149847074911565324966995920 x^{24} + (-24168348772820935520184035920a + 41230922056027926556297805624 )x^{23} + -312506954239922769581635894616a - 339390478862928588164035974616 x^{22} + -185487449806854659559621856944a - 270782652359702194557015701616 x^{21} + -245655996529070165300524532480a - 531079846582698855022666844360 x^{20} + 63483655948842371519434775532a - 51823240499437544350654506928 x^{19} + (-354734046146016874635591406960a + 340038372128529730360303221552 )x^{18} + 297099492198258171038459611016a - 494506587865591430113861132504 x^{17} + -254182185920199478899057476584a - 180052123397868474735704163056 x^{16} + (395886081992328238968986792432a + 561960342044501069744793627648 )x^{15} + 188585248208048782119482434796a - 249446195647312266667457198480 x^{14} + (280130726656965803703605250624a + 316664654899588873058072909000 )x^{13} + 76230295450432543702782102280a - 38202526431806009525366244228 x^{12} + (-555911618841308115170332727032a + 97474852973962644318654627728 )x^{11} + 516225148946618853839628487680a - 456450440236673301322367768064 x^{10} + (-8226777391396429009629342424a + 295547267979957565372647446096 )x^{9} + -577619560943690871065731266048a - 571428363621636420396938395216 x^{8} + -410393522843401209316937953168a - 148452829724694488794853137512 x^{7} + 433142243349374439466199123052a - 243789443353490622613347650352 x^{6} + (35919445943926690883503143712a + 86008546137199162516141434088 )x^{5} + -381901570200305986902082593544a - 351137999742521640425151463520 x^{4} + 126973504161497536703945136384a - 376967040354737277080503426336 x^{3} + 354129738308038023620309083936a - 259750139956087562785511938656 x^{2} + (92483874591278850372850448632a + 137220492535377483039234009136 )x - 545561996050354557505225756888a - 250013715322298996354606223266 \)
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