← Back to 2.1.2.2a1.2

ex.24.10.1.131_259_387.a

Base Field
\(F = \) 2.1.2.2a1.2 \( = \mathbb{Q}_{ 2 }(a) = \mathbb{Q}_{ 2 }[x] / (x^{2} + 2 x + 3\cdot 2 )\) 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 \(x^{2} + ((70425033346012744527594622521a - 70425033346012744527594622521)b^{4} )x + b \)

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

Inertia Polynomial

The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + 105705502971403697356376867080a - 74728410457774133010420432664 x^{47} + 436883239401573477870408327340a - 152643365165534702940751489736 x^{46} + -165851358931141178762453455636a - 140940252320090649384705599328 x^{45} + (600801956151711991987416499368a + 437506571138585902262713947444 )x^{44} + (-460698536908552585375764336216a + 141868326816605128062698729648 )x^{43} + (151307241810369035667853017672a + 427275936799125349764201641516 )x^{42} + (230895374201761812446232714360a + 373459811630219261674707133688 )x^{41} + 449959992213313980748071872328a - 558801209909262500435649034932 x^{40} + (41614216486425114896758848200a + 538551453732245283690062406816 )x^{39} + -89848815251440877144448876288a - 476725017646935337984047350376 x^{38} + 474303033969590148002246314380a - 503023716754310444287276658576 x^{37} + 194184795126857154242355694100a - 598954902960171028222417589668 x^{36} + (-520710212748639058931888764704a + 300166351636841112487003446432 )x^{35} + 333543190660800134373554575844a - 234538930837016473511395884176 x^{34} + -468683220127483234980608499272a - 316540188168501592970220904896 x^{33} + (384176218894990187678309448184a + 443597198670318857200858561224 )x^{32} + (-233836822235128790058553938016a + 425729476366132235065595236864 )x^{31} + -324456559522205438679032059328a - 38706846352081574051690538832 x^{30} + (-150732821094251819240562006472a + 184293851078813523218943553256 )x^{29} + -121247892604162639976091449864a - 611873510797490085710697784744 x^{28} + (591635124274323901365055107504a + 577599916996519384432232812720 )x^{27} + (-397864494802939363940910691972a + 524172932037710304127322961368 )x^{26} + (275131746674423298536170086880a + 54434122984719938235672149144 )x^{25} + (-444287161812308617624172410194a + 152376037208047220105678806388 )x^{24} + (-355199035431372509407045642808a + 367290341482739267451977343840 )x^{23} + (-559656710129390336562740130176a + 8814015501394743958133859608 )x^{22} + 416338453808835677706041987568a - 34884721954729940591951437208 x^{21} + (-597449995104859443147539644612a + 291526093113613005788752215880 )x^{20} + (311503571247427155629960623328a + 454435960984000611586813224880 )x^{19} + (2555209883056514930261814028a + 591566923524795050004039603928 )x^{18} + 314213577458257863631020387240a - 69675863021673867009691409008 x^{17} + -181304786663457077319168732228a - 16659376364042687370460881512 x^{16} + 9899524262319787107273062896a - 349255209221460402557749012736 x^{15} + (388907299982902323241310935504a + 417149934605416483728553767552 )x^{14} + -420171934990972092985528111136a - 50153987665143496675141366232 x^{13} + (64101536621033548856966078520a + 301139162806446823485923332144 )x^{12} + (416927371633957349248020982128a + 533346843966184716187143029728 )x^{11} + -256678083751410070404001504232a - 355696548915664759590652110680 x^{10} + (-543410768030558182413961429824a + 266141523699312859423310637648 )x^{9} + (-215077994241385434887357217608a + 460794372469857685620476403488 )x^{8} + -75384156332423108650827926736a - 89588338731667476540570400880 x^{7} + 244551389977221797868549691504a - 583732606365841407111675689280 x^{6} + -562842083675496464304161936520a - 57887705572167509385346882976 x^{5} + 610034693566363381319887487256a - 587930477621630788239723424928 x^{4} + (322882488357686378707346163664a + 533210021737552848904496005280 )x^{3} + -623423597525672190765048126008a - 359225121320133442402328379656 x^{2} + -169309872428219408163687307656a - 198148835069858953413879375216 x - 16200465112164374647799510054a - 451929952001252118791818179738 \)
← Back to 2.1.2.2a1.2 Summary