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ex.24.10.1.127_255_383.a

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 \(x^{2} + ((a + 1)b^{4} )x + 3b \)

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

Inertia Polynomial

The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + 339916984295338558880293426472a - 474712376197493543101902001160 x^{47} + 567996277027897430136758710908a - 283366008291683089496020223636 x^{46} + -10316241910678859513314784764a - 542742066973148913334185340352 x^{45} + 399215424626141654571953340960a - 225431478887856686874510302000 x^{44} + (50508239204968216466039526496a + 596977647742394757870254274064 )x^{43} + (-292645571939560317877978647296a + 420653825137874001220619046044 )x^{42} + (-575223852631764472914934889632a + 418129188592500717397234473896 )x^{41} + 276280038679644005588174109844a - 41624516952789129834248619572 x^{40} + -185552364114872755083963127848a - 178906476973226752840387021520 x^{39} + 617377992272847104657643169304a - 237826296980899632648130705352 x^{38} + -312808358436018840172312298244a - 445070314731942802938027217448 x^{37} + (-174128074476396331083103503184a + 92950731733562649102087942116 )x^{36} + -339405125445011125394412479816a - 548312868938886647232764460144 x^{35} + (206146967364101186902286043296a + 147631090520002207475782290808 )x^{34} + -403927145695065781735452382152a - 124086733798614251719089540896 x^{33} + (557647250455746748336248953376a + 401132601816264059955750490384 )x^{32} + (511663186461641721558650765824a + 62649808151016309141257652736 )x^{31} + (24952941937773412564406038520a + 299918991874798489920566494768 )x^{30} + 21686393197762967676973649048a - 379902239546799333251634552064 x^{29} + -484786103596101953941406367140a - 324447955053097871634403449160 x^{28} + -494319184979354438287536482336a - 42918620756959897128673907440 x^{27} + -402728327778634542751363087240a - 348671704733395252823720760344 x^{26} + (408592810070506350223757546768a + 69450603716458768861483541208 )x^{25} + -277123456933369194982862253522a - 570235518464681833027508519072 x^{24} + 359842217537229066655875295928a - 359456463183956040987804091984 x^{23} + 429483233890658849316947401444a - 613046553727207053133734208136 x^{22} + (-283521356456900141462157249192a + 286926264301457197210915460168 )x^{21} + (9861197659010740599739728240a + 306985587501639757120703779056 )x^{20} + (-477253831721765029288599316128a + 103303489465479002408094790192 )x^{19} + 128980884117464086753436444924a - 145691993034135619755519205808 x^{18} + (11733461442012631936037064248a + 36151987962502727067275945344 )x^{17} + -397336022270721273921879019348a - 129842302173227226399372815352 x^{16} + -149895047115953749267846310816a - 114861464034396302985931937056 x^{15} + -10112711211624764664361866904a - 320394212540311363665516819632 x^{14} + (-222663014049119526435318464080a + 543554316729318675794490748568 )x^{13} + 330143118396944596548828588112a - 513562412866807987962379105912 x^{12} + (-544349938289518643350274477488a + 10805520246823126017698328400 )x^{11} + -27025689890525383446332151792a - 453752204243172318301638448080 x^{10} + 75630835860932397664135646432a - 445795772555510629870313120688 x^{9} + -607363197155735902189609611776a - 539609818058238503720246465248 x^{8} + -615464392182736673736645028624a - 483573696545878881975918299472 x^{7} + (220043009511525802657451101856a + 585434526534179815027584551840 )x^{6} + (-412561101170409486366773672336a + 184176348639377676519907169360 )x^{5} + (-306625900844524136121120692752a + 571467212783930001077455841208 )x^{4} + (-194075864775400781276120607024a + 385987058671729248117181618912 )x^{3} + -162222814513542256274271333768a - 164883186711959778937423162896 x^{2} + (605986929713949932280254789336a + 419662782472070576717901487104 )x - 222867875741636902796342672330a + 36070150276329478243135441494 \)
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