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ex.24.10.1.131_259_387.b

Base Field
\(F = \) 2.1.2.2a1.1 \( = \mathbb{Q}_{ 2 }(a) = \mathbb{Q}_{ 2 }[x] / (x^{2} + 2 x + 2 )\) 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 \(x^{2} + ((a + 1)b^{4} )x + b \)

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

Inertia Polynomial

The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + (-474211381130228606402700099640a + 362793698867465405522521801784 )x^{47} + -70069214467356094160629037152a - 109320880266797021617081254620 x^{46} + (148128963546458508178303537740a + 48784072908360863511358880960 )x^{45} + (581955994233534075322552983124a + 161465215928326563198199882784 )x^{44} + -482337875972178654188707121144a - 249042903185540944220350198048 x^{43} + (495306090034905991770918293704a + 302934702842968692923487693068 )x^{42} + (145763665448206215769089145304a + 501514832076025044227498133512 )x^{41} + (311033666614992540424567414236a + 321184665735665016666723426292 )x^{40} + (82953436663871613331138523800a + 512094750010335344460825477888 )x^{39} + (139306735000450253378501564096a + 218726572218324336498077084880 )x^{38} + (-65382880207350398040985923548a + 347694719261930171007060512760 )x^{37} + -332569318388746014317262423280a - 196763792578962136184506285492 x^{36} + (383580884232283454703265226816a + 98765293155339162816864765920 )x^{35} + (-152115742021517660641079938452a + 557832956962409180553529693368 )x^{34} + 522835884035520306121398604216a - 123578567694525428276469296832 x^{33} + (-150020339737856447371852224308a + 590810732510584285054752883376 )x^{32} + -414569767795591939087688002976a - 380332455542195152872339139680 x^{31} + 385984230129224997223530780504a - 322769612560474288377020451008 x^{30} + (471756250845420600310783492800a + 446011925118035496358333543568 )x^{29} + (-540124656710492437642453883956a + 286472387708819189682479626008 )x^{28} + (183973631912676422145565579120a + 34417698869969878473245001520 )x^{27} + (404032845784968529738205025200a + 143180062805151366067778330344 )x^{26} + (72057259330524320499239927824a + 431137574602092072460250636000 )x^{25} + (-55168204247340272922763051850a + 158266539581838089357904189772 )x^{24} + -333730105635631193020488236648a - 541745975584934033004071412384 x^{23} + -383166387066928058146812548940a - 580456288416267055854665283096 x^{22} + (-45476160408836189938363390304a + 226411533669089239347211772888 )x^{21} + -166195554380520796169170759088a - 167609209013115117648938815432 x^{20} + -420459902637967213173618885264a - 589231925165741363607326200240 x^{19} + -143558627108651437170189715908a - 522606227099987067458256151768 x^{18} + -377320744438193619499693005384a - 377225632424625542974586181920 x^{17} + (-521909551903578721863539147620a + 417002210954676684738298856464 )x^{16} + (-471366715274863846643279356496a + 545090214027492526051581348256 )x^{15} + (-273419062266886717915713845360a + 183341665100351839527593460608 )x^{14} + (375339955633533441968966741288a + 324909101549867765680432027960 )x^{13} + 56224375831114174962213842720a - 609806292816914505780744902680 x^{12} + (453132417675846519067328800624a + 405640787761469118368800356608 )x^{11} + -311975882646335785770841055120a - 437793830404705822195863781512 x^{10} + (473942673969758908853288128624a + 524914083840396940451978852528 )x^{9} + (-134062172670764629386573844208a + 64584202700655571687104724616 )x^{8} + (220030375901323470339861817616a + 485310063363472474825691013936 )x^{7} + -128727492034509230548705981104a - 30624601712783613783355540320 x^{6} + (-87594150334164617600942949792a + 41622588072604383662637198720 )x^{5} + -153421566962455230334337239608a - 543121478525361804004596984088 x^{4} + -400829190259126952459523499504a - 121429790753088627087837420704 x^{3} + (330563238589243456442128132536a + 97124056903102780122050361152 )x^{2} + -17938725730635766630837741648a - 370573665119126231164802754624 x + 328265065177967481319805615970a + 146831232091989604993960837402 \)
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