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ex.24.10.1.31_63_95.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 \(x^{2} + 253530120045645880299340641075b^{3} x + (253530120045645880299340641075\mu_3 + 253530120045645880299340641075)b \)

Underlying Character

Character \(\chi^A:\mathcal O_K^\times \to \mathbb C^\times\) with the following properties:

Order
4
Conductor exponent
20
Values on generators of \((\mathcal{O}_K/\mathfrak p^{ 20 })^\times/U_{\mathfrak{p}^{ 20 } }\) :
\(\begin{array}{l} \chi^A\left(\mu_3c + 1 \right) &= i^{ 1 } \\ \chi^A\left(((4\mu_3 + 4)b^{2} - 2a\cdot \mu_3)c + 4\mu_3 + 1 \right) &= i^{ 2 } \\ \chi^A\left(((2a + 4)\mu_3 + (2a + 2))b\cdot c + 4\mu_3b^{2} + (4\mu_3 + (2a + 4))b + (-3a + 2)\mu_3 - a - 3 \right) &= i^{ 0 } \\ \chi^A\left((3b - 2\mu_3)c + a\cdot b + 1 \right) &= i^{ 2 } \\ \chi^A\left((\mu_3b^{2} + (2\mu_3 + 2)b + 4)c + 2a\cdot \mu_3b^{2} + (2a\cdot \mu_3 + 2a)\cdot b + 1 \right) &= i^{ 0 } \\ \chi^A\left(((-2\mu_3 + (2a - 2))b^{2} + (\mu_3 + (2a - 1))b + ((-3a + 4)\mu_3 - 2a - 1))c + ((a - 2)\mu_3 + (a + 2))b^{2} + ((2a - 1)\mu_3 + (3a - 3))b + (-2a - 2)\mu_3 + a - 3 \right) &= i^{ 2 } \\ \chi^A\left(((2a + 4)\mu_3b^{2} + 2a\cdot b + (4a\cdot \mu_3 + 4a))\cdot c + ((2a + 2)\mu_3 + 4)b^{2} + (-\mu_3 + 4)b + (-2a + 2)\mu_3 + 2a + 3 \right) &= i^{ 2 } \\ \chi^A\left((((3a + 4)\mu_3 + (3a + 2))b^{2} + (3a\cdot \mu_3 + (a - 2))b + ((2a - 2)\mu_3 - a + 2))c + 4b^{2} + (2a - 3)\mu_3b + (-2a - 2)\mu_3 + 2a - 3 \right) &= i^{ 0 } \\ \chi^A\left((-2b^{2} - 2\mu_3b - 2a\cdot \mu_3 - 2a)\cdot c + 3b^{2} + (2a + 3)\mu_3b + (-2a - 3)\mu_3 - 2a - 3 \right) &= i^{ 0 } \\ \chi^A\left((((2a + 4)\mu_3 + 2a)\cdot b^{2} + (3a + 2)b + (-3a + 2)\mu_3)c + (3a - 3)b + (-a + 3)\mu_3 \right) &= i^{ 0 } \\ \chi^A\left((((3a + 4)\mu_3 + (a + 2))b^{2} + (2\mu_3 + (3a - 2))b + 2\mu_3)c + (2\mu_3 - 2)b^{2} - \mu_3b + 2\mu_3 + 2a - 1 \right) &= i^{ 0 } \\ \chi^A\left(((-3\mu_3 - 3)b - 2)c + (3a\cdot \mu_3 + 3a)\cdot b + 1 \right) &= i^{ 0 } \\ \chi^A\left(((2a\cdot \mu_3 + 2a)\cdot b^{2} - 2\mu_3)c + (4\mu_3 + 4)b^{2} + 4a\cdot \mu_3 + 1 \right) &= i^{ 2 } \\ \chi^A\left(((-\mu_3 - 1)b^{2} - 2b + 4\mu_3)c + (2a\cdot \mu_3 + 2a)\cdot b^{2} + 2a\cdot b + 1 \right) &= i^{ 3 } \end{array} \)

Inertia Polynomial

The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + (487301373115530902857108970204a + 137716051067326585018195009080 )x^{47} + (382824965198134053091688729996a + 16409394188067240924491756808 )x^{46} + -453991431488653042061393742792a - 613461535404423683736728430672 x^{45} + 381031773277107274103966073632a - 339302074784620417987970148312 x^{44} + (-581761333605420620246607991856a + 408432901214717462222741821516 )x^{43} + 568704757826656644042228755344a - 323854219293081834959837516536 x^{42} + (-313012115246113907445352505308a + 285989897587376085205309968824 )x^{41} + (-8298507429591724528964157840a + 331362933926415413036455822760 )x^{40} + -82917710027680250680812121896a - 452399148201496545459865665984 x^{39} + (-463060332550185032652277994912a + 212777282836285172354594768596 )x^{38} + (-338052857055237247754100291724a + 248698216685716947943102005896 )x^{37} + (529186903182923929830231504716a + 55025075708064022074088367200 )x^{36} + (457024855836997907541017726872a + 421276244258255100138840925184 )x^{35} + (140837582339169724232678518304a + 248267143362793666857931611680 )x^{34} + (-227422141061353923993570666776a + 383306376545538621366051163288 )x^{33} + (-133123205138414902487795960a + 509471915243080164633248979168 )x^{32} + (-143261537065530735834974336292a + 545946050985865221048525338096 )x^{31} + (-50896957808427593776965411016a + 86106256882293369832179902300 )x^{30} + (-92213342103299220137902456052a + 461348565818967757210964793600 )x^{29} + 97866190761168593005330139016a - 459782176957567722303579475912 x^{28} + 598332910867771525109079177312a - 167027585860417274560329268832 x^{27} + -249143543978564065808285477272a - 81084289642784098753122785872 x^{26} + -216949557428372973045015120984a - 133761121451947130525861168928 x^{25} + -360826273956605321407512740266a - 575366130964488241975230641488 x^{24} + (304494304019409384455774814192a + 60158308833030974840290731000 )x^{23} + 172884306827313735431630920552a - 229046623328264083435503263800 x^{22} + (620274138803201641720574800512a + 46123055284366037483558622800 )x^{21} + (517077297218565078887703680688a + 126037302522678617237182692360 )x^{20} + 26940375682586136586466340076a - 292228264387468276209583968080 x^{19} + (366820897795008869766723691760a + 541177193052615867709574334096 )x^{18} + -590734417308227334942437193752a - 120688774239789128096805226984 x^{17} + (297200093087093132765416625816a + 567789334588311182451362209488 )x^{16} + 214666136975739873246002801200a - 561273764112543482477521504544 x^{15} + 618033667097605027441652377420a - 292599632167757277850553842352 x^{14} + (-602084233450171750313620975856a + 335305254972970327311508762552 )x^{13} + -275274905997849043496850416024a - 427133976606974246117723523780 x^{12} + (436932160930616323239334826600a + 509543967128336848530976047888 )x^{11} + 600438459721116513266430512352a - 299982456273481272514666068992 x^{10} + 472632807837497773657157531272a - 21215968944073203576768435824 x^{9} + (205294527042848136599223402224a + 127014406468316592991009194848 )x^{8} + (-117741472213149327088843601680a + 187746603734137194392387476920 )x^{7} + -163069587595615373480955694916a - 537943201484751324737746246384 x^{6} + (5603762857608320572587891264a + 199658632059340185727613142712 )x^{5} + (-371228608478801851621974259848a + 6328939066036958748346986400 )x^{4} + (299326685675320012420161209952a + 18508572767073616007027266592 )x^{3} + 445438562395222005618474360000a - 281903997780476080906515273216 x^{2} + (-230288668517801423664174385192a + 187696947807239266203151785520 )x + 318080364151967633768637439368a + 90747751837498917886791312342 \)
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