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ex.24.10.1.127_255_383.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 \(x^{2} - b^{4} x + b \)

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

Inertia Polynomial

The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + (426620235925232049872189753456a + 245824284014678529528181989112 )x^{47} + (-429521955197202034796591961980a + 106822699329578425778084185920 )x^{46} + (135523278337229032528285318348a + 190969918254878779934966850896 )x^{45} + (83194550973427025402196203008a + 514292011385110863850478148044 )x^{44} + 238415425241909275540352164352a - 76431946919483846125581380384 x^{43} + (62911584896958870380634993244a + 44893978669737682847859032620 )x^{42} + (543781741514005233884997318648a + 99838482300256594540851112464 )x^{41} + -223268139045966576827502718760a - 557710567138820017809493783180 x^{40} + (209414125232391992979239084888a + 241992964841516557762375364576 )x^{39} + -13962119925331977407288494548a - 3765854798250930367487218928 x^{38} + -373803571645110209963089129748a - 209572752638694189373234598688 x^{37} + 11354155231106482868682683292a - 36172653798779952993615559868 x^{36} + (-529128052465664264099007276328a + 228963388609756066390456016864 )x^{35} + -493719580782487325427099387692a - 297279411014732434631247050488 x^{34} + 15527681514328653891835322584a - 112275117488553225099479146144 x^{33} + -308647149812949918090662770956a - 155776034617468996664719258816 x^{32} + (-260468652800562368870080165664a + 143531547364789235915112028192 )x^{31} + 269329521039580835813191395584a - 60156460674480956547616518416 x^{30} + 154119415583470496411786781592a - 626258729645295464041200993176 x^{29} + -361355340811384966854742262556a - 161130292133471940452721949472 x^{28} + (276523588230711148448025428864a + 37352979112184503331463571408 )x^{27} + 17640173326425574443481715492a - 541819073510761309262170478816 x^{26} + (148002388691200265663186218408a + 253509749492241750777694227216 )x^{25} + (-305073799440770502170399779946a + 562638880160439803174101611800 )x^{24} + (413600441773822897931709319048a + 496186529609616599315712956864 )x^{23} + 29422557965426662521691978400a - 61738478011708301496382015496 x^{22} + -12297722964080960784162995600a - 289447833382683367392480381768 x^{21} + -584301799806562677893763426172a - 561061250481189779236648943280 x^{20} + (-215407973560071334757917244448a + 234605779182480821656622110256 )x^{19} + -127589400731453167238947529428a - 231901516148808914778322295352 x^{18} + 445463466425858189838535327312a - 239835486045217400695364499744 x^{17} + (-272587185285682398610082506188a + 446705701101226267271505234592 )x^{16} + -343097284649178248347348259168a - 608613449903319125372972318912 x^{15} + -444993760322396333555469876216a - 594507094108507209743568751128 x^{14} + 80978521806576789101688962080a - 26175963306454620404462485512 x^{13} + -249810407480909971064492030072a - 611158427293763701250261781136 x^{12} + (452108564405687500306228338640a + 627657660794277465849212672592 )x^{11} + 438671952138590963597944334848a - 463623157270669070925498283304 x^{10} + (-175557213612476632239922713408a + 159934287646059900175131657168 )x^{9} + 493820808522565480989848342352a - 32579202987036172347394399400 x^{8} + (-4489489106748016792695252000a + 326207663224621463959149718288 )x^{7} + -15194427605162251494002739168a - 532337761484759610695054909824 x^{6} + 134595620931646922147885571176a - 363562359475976139298331326352 x^{5} + -394380710207767173234542981616a - 133871846087684193319832719832 x^{4} + 573005894568101073501523671312a - 180635642668078657825475037664 x^{3} + 509717270127468639293612999728a - 301685251380335609554830912808 x^{2} + (580366281489760436362751014096a + 528069282754202584244691600464 )x + 318440772479611875263311235112a - 141388943318776317288198314362 \)
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