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ex.24.10.1.131_259_387.c

Base Field
\(F = \) 2.1.2.3a1.2 \( = \mathbb{Q}_{ 2 }(a) = \mathbb{Q}_{ 2 }[x] / (x^{2} + 5\cdot 2 )\) 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 \(x^{2} + 253530120045645880299340641075b^{4} x + 3b \)

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} + 190187233347240206072886582352a - 592791236836087080759662173608 x^{47} + 562219194495633516306209261300a - 471767878939300859249397388080 x^{46} + -479739920910291650382289422332a - 17964691472033347947119093712 x^{45} + -283573607993738205760183276308a - 626511614242397698228803129396 x^{44} + (576836586678971128633279172552a + 232471871931884017931251504528 )x^{43} + (-273662645162530762916354750532a + 514546788221693164859022642748 )x^{42} + (234770815220384483477520743296a + 288243603363981547557731789848 )x^{41} + 169186573846578001071853591248a - 273567510447701570747916058812 x^{40} + 257207135555888236677461003432a - 561241920034583446225706342112 x^{39} + (-401950754096364606354492325604a + 321755903687433542019539852904 )x^{38} + (-29657376844805152927639127700a + 581833861269444420797057651672 )x^{37} + -68208534442164402602593846776a - 106131860784147609363852817172 x^{36} + (594944337754619187747478965680a + 397848350707879124798884963120 )x^{35} + -331243499282217371259552843260a - 49260169687366413783086714488 x^{34} + (628686641113123557915756404544a + 482442333789500368579520729656 )x^{33} + (228184523813439658892633199524a + 91622183574977489911880577424 )x^{32} + (-433340759410039150712877833088a + 484926981182051277250014330080 )x^{31} + (434875842239039184316294995896a + 238506058589492225228456605312 )x^{30} + -210593196511235991506514175488a - 360610036176842065302006678632 x^{29} + (128348706586493428977139574748a + 290732396466013503807467669912 )x^{28} + -225673663745675544497137638512a - 132778347773010529755338892864 x^{27} + (577410736707111564129994188860a + 366897922203599448071132559896 )x^{26} + -51205677852531784952179263496a - 151217890744614531505842428760 x^{25} + (7000727864283207395990042546a + 167341015754752193971274751708 )x^{24} + (130074281173490308998927231720a + 631816885226271284984300795664 )x^{23} + (-137080666798058090590479522984a + 377691707545894998101541126376 )x^{22} + (148308139009071219686646392776a + 362298551103227289745576492568 )x^{21} + 338820806964057727908986814292a - 297714425680759304652924012704 x^{20} + (40587095781802750601905742656a + 341996888422227607273489763632 )x^{19} + (-502735403158501674300169480052a + 395409369838979866896482830960 )x^{18} + (-63599292887268278984312579656a + 203410230710346696729481986624 )x^{17} + (76573208268697718804322244972a + 623331610322024725111808440472 )x^{16} + -455825256252560952437342709280a - 365895310613654044669597446784 x^{15} + (570532466412174097799825457176a + 15226212881004902491194218552 )x^{14} + -142013995426688720497735812960a - 63489120383572855659154076168 x^{13} + (616733047319281635529915052728a + 553236663113883191583949663048 )x^{12} + 61329122619920350481940405920a - 604212273738408090090901241632 x^{11} + (225571062143579448193997549992a + 414458681954127908491938724408 )x^{10} + (587432548605016614365213469736a + 99392914107919267738421583312 )x^{9} + 463871366548798480242147471784a - 369397596556574246217114159416 x^{8} + -173131905107712910354527207168a - 211967006115738028999022889392 x^{7} + 601048290630919905919188125776a - 344879831505710487779990681472 x^{6} + 613773054648667862538806882520a - 120741909374164629982585893808 x^{5} + -487200762767636348519910588816a - 452862795734768479084695496840 x^{4} + 570650498018498340273588576944a - 546913210641615954644644502912 x^{3} + (-128798589148014927556379892624a + 40320894683687970853554749672 )x^{2} + (30670958466869472454510560856a + 298227901822498332073038516864 )x + 208381203938265656949856422060a - 239447910363676741156423256382 \)
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