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ex.24.10.1.31_63_95.b

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 \(x^{2} - b^{3} x + (-\mu_3 - 1)b \)

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

Inertia Polynomial

The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + (214879693557126984625408419948a + 494769446000837899851484609232 )x^{47} + (207362959108520734138641932028a + 484438744172221992900959319224 )x^{46} + (37867759325875694572128059280a + 285684390436829774800069410776 )x^{45} + (-251049746671689782024723100612a + 87610680298788572995657212936 )x^{44} + (104308309287407432739260420136a + 210181153379145570073566481628 )x^{43} + (553895011122001448866719457832a + 565345414009849934880064739736 )x^{42} + (131375336238934726394627003624a + 533662623124560176839233881640 )x^{41} + (-98786023591336982090460743272a + 314107230752450232576898918672 )x^{40} + (-272821962252089556083935749944a + 494292152165904587307077425776 )x^{39} + 590066328948146296738542858360a - 457296316589062547614497198764 x^{38} + (-154120567956272127164015430884a + 428066167694906770520822756024 )x^{37} + (375317535227655778758642782560a + 431592889279900225617751761064 )x^{36} + 251960294580132734651176843832a - 371538346198466829244591729936 x^{35} + (375833873252319788694387201856a + 605875631583929592397688474320 )x^{34} + (620084699669180248258788984056a + 533804291537271696542464978968 )x^{33} + (41465740390145994995819032720a + 493615455450329123264001496024 )x^{32} + -270910041950768963769686609776a - 16348979367328945597885496416 x^{31} + (-381718532093526815517845333520a + 274803858796062136313016912220 )x^{30} + -526172201841593398574198192140a - 225911878446414509358379843072 x^{29} + (245466663187485559646812265744a + 454824604531217593747065913720 )x^{28} + -585093194028394267232320396544a - 603434268192493399803411399920 x^{27} + -16440256983504585288060755864a - 436150849292810157320282112512 x^{26} + (40774573269013414654166084448a + 483450156942096304113116219584 )x^{25} + (-284389883277697910031938433010a + 111656105131231478719798225664 )x^{24} + (401300877094237183825358189696a + 130149474992692960883827910296 )x^{23} + 397789122160719493958393772648a - 533456045867362889577971576360 x^{22} + (257729190110524466664276802472a + 287598456333168938346305954208 )x^{21} + (-188557768818356734664147619680a + 368232164159807563114275729568 )x^{20} + 159637232750948498621810282972a - 264790986846078675771263981056 x^{19} + (168768448249454978565325392840a + 588324981660887607556294577296 )x^{18} + (-368677937030887491227875369928a + 306217133569628570472534907072 )x^{17} + 368499588697079908955613290912a - 159031191515429442195888151712 x^{16} + (111544083404396064014423662912a + 592108893643804174857997866944 )x^{15} + 80612448635505831888057655756a - 221061312866638905802437538976 x^{14} + -243590887284015118623727864024a - 223309363839503468751491787512 x^{13} + (-65475195875445158427698984784a + 111154724991374929903773038980 )x^{12} + (299417545684036180262996087336a + 280608292759708408761161246192 )x^{11} + (-599320261734414902974719992448a + 551151861785056455299470176832 )x^{10} + (345517330416053205741165729896a + 570690913065088758231038064528 )x^{9} + (-254381223680861038674180916120a + 617301241634216717704586255984 )x^{8} + (223882756575492000474357889648a + 366572424832801654842326357744 )x^{7} + 553456345152723778253718559148a - 340619497229114769910532580192 x^{6} + 524660970431222882987647257504a - 545246994133575078966920847832 x^{5} + -117647253921747381113839897960a - 504474206744623180041896755088 x^{4} + -387279343022001691623210522048a - 27551336889941066345407938560 x^{3} + (108438752063070140113912698592a + 237490171247981359608878200448 )x^{2} + 518366401446447971452174861688a - 95028752031787943945377964000 x - 610717026338875645741759719408a - 299618219142977836070671745218 \)
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