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ex.24.10.1.131_259_387.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 \(x^{2} - b^{4} x + 3b \)

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

Inertia Polynomial

The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + -625634798739121866452036941616a - 608134777612697930571714073608 x^{47} + (-409895737732771325751005033692a + 245996136016395631709190570400 )x^{46} + (195300605888886766404019308804a + 509240546542039168613930661040 )x^{45} + (-96715491685489564394710573236a + 559999506352355735391702745484 )x^{44} + -294092962293108190631302912472a - 86744161904041830066635988976 x^{43} + 58417311703819898787754847596a - 134202476101372608252681777140 x^{42} + 587839476569781070447311085456a - 584044829077002218879361995336 x^{41} + -342131468593630487928909805104a - 485566094133843744281689417708 x^{40} + -111826599986569084359571001016a - 339493640388805092162359506144 x^{39} + (16460024901396394961586198508a + 151492831069919449133762397512 )x^{38} + -196918614319029277559577156228a - 269635570300090029062193256616 x^{37} + (290207106443926942095741999528a + 296486062773928758640721005324 )x^{36} + -603734431831209259736694204720a - 305971372245426789128506049872 x^{35} + (582508176580502752286207257508a + 616479480584069817352573237960 )x^{34} + (350283866250801430132101311840a + 359549975101000310935702412120 )x^{33} + -381933143176532943691433600860a - 103712199970049196021656241200 x^{32} + 567497868077176448990662233728a - 16753023489615081455097565984 x^{31} + (-56484610238619790673200105128a + 279885629659869545695293710336 )x^{30} + (246868901411947451772335902288a + 610468012594147398761365319320 )x^{29} + 33714352937974769881069658764a - 353862597837883618965040044392 x^{28} + (209077816326299384197800547536a + 466935424940149764156046950848 )x^{27} + -390928082875059970027800889492a - 21590248613978018314726302296 x^{26} + (402351156613341283742506736440a + 176853242091950521375968146472 )x^{25} + (-618313968082772330694651733886a + 171794654228975480475452345836 )x^{24} + (-469835285588324266456954476280a + 362821702558932410228880632080 )x^{23} + (17331465707621452607686287528a + 336210785108806632949278053768 )x^{22} + -321711767222904093614006786136a - 300962500158963451702812065416 x^{21} + (452964205100479227825113739636a + 215488831737804576908773709504 )x^{20} + (-633011569402898923551818836032a + 111449234081964942803299902640 )x^{19} + -577758145661787148491831938980a - 315367015170153684588140391248 x^{18} + (335949282590930334276739777272a + 456229586111242279518036324768 )x^{17} + -555629289919971574454945095108a - 103124145718045401602360572072 x^{16} + (533975474152088913179313184928a + 75752253516686002091416492800 )x^{15} + (-217066572536529090324805037864a + 278155768391038172140277898872 )x^{14} + 20710429696097400087636665376a - 6009228682983173504032929672 x^{13} + (389411786596387097880683682104a + 491866255566415692349823447976 )x^{12} + (135268215862956775355797580064a + 100010680446464957571927788064 )x^{11} + (-427775614033634229075030515256a + 360102811177085055293586612504 )x^{10} + 62699103084440675539086888968a - 470139476356669711767073289328 x^{9} + (-612665970066038221177883132568a + 412494694945626065343956703400 )x^{8} + -517235844907766795589574018688a - 99800165181703742828229516784 x^{7} + -404130724779662733055499821744a - 539408191939598165942467069952 x^{6} + -151078159349741530674317030472a - 190701440079871476171147081936 x^{5} + (27814763812134895468590133488a + 517678547455331528065556664824 )x^{4} + (446669216988049217161635239728a + 618523013377054658762899073280 )x^{3} + 128788905850955498439111789920a - 252251743663315116032508062520 x^{2} + (239759112538198973397081877592a + 167261230493567258747277349120 )x - 195284991296486276897293505316a + 119700783776261054593416738154 \)
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