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ex.24.10.1.33_67_101.a

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 \(x^{2} + 253530120045645880299340641075b^{3} x + (63382530011411470074835160269a\cdot b + 253530120045645880299340641075)b \)

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^{ 0 } \\ \chi^A\left(((2a\cdot \mu_3 - 2)b^{2} + 2\mu_3b + 4)c + ((2a + 4)\mu_3 + (2a + 2))b^{2} + ((2a + 3)\mu_3 + (2a + 3))b + 4\mu_3 + 4a - 1 \right) &= i^{ 0 } \\ \chi^A\left(((2a + 4)\mu_3b^{2} + (4a - 2)\mu_3)c + 4\mu_3b^{2} + 4a\cdot \mu_3 + 1 \right) &= i^{ 0 } \\ \chi^A\left((((a + 2)\mu_3 + (2a - 1))b^{2} + (a - 1)\mu_3b + ((-2a - 2)\mu_3 + (4a + 2)))c + (3\mu_3 + (3a - 2))b^{2} + (\mu_3 + (a + 1))b + (2a + 2)\mu_3 + 3a + 1 \right) &= i^{ 2 } \\ \chi^A\left(((2a + 1)b^{2} + (2a + 2)b)\cdot c + (2a + 4)b^{2} + 2a\cdot b + 1 \right) &= i^{ 0 } \\ \chi^A\left((((3a - 1)\mu_3 + (3a - 2))b^{2} + (3a + 3)\mu_3b + ((2a - 3)\mu_3 - 3a + 4))c + ((2a + 3)\mu_3 + (3a + 4))b^{2} + ((2a + 3)\mu_3 + 3a)\cdot b + 3\mu_3 + 4 \right) &= i^{ 0 } \\ \chi^A\left(((-2\mu_3 + (2a - 2))b^{2} + ((a + 4)\mu_3 + 3a)\cdot b + ((-2a + 4)\mu_3 - a + 4))c + ((a + 4)\mu_3 + (3a + 2))b^{2} + (\mu_3 + 1)b + (2a + 2)\mu_3 + 2a - 3 \right) &= i^{ 2 } \\ \chi^A\left(((3a\cdot \mu_3 + 4)b^{2} + (-2\mu_3 + (3a - 2))b + ((-a + 4)\mu_3 + (2a - 2)))c + ((3a + 2)\mu_3 + (2a + 4))b^{2} + ((3a + 1)\mu_3 + 4)b + (3a + 3)\mu_3 + 4a \right) &= i^{ 2 } \\ \chi^A\left((((a + 4)\mu_3 + (2a + 2))b^{2} + (2a\cdot \mu_3 + 4)b + (2a\cdot \mu_3 + 4a))\cdot c + (2\mu_3 + 2a)\cdot b^{2} + (-2\mu_3 + 2)b + (3a - 2)\mu_3 + a - 3 \right) &= i^{ 2 } \\ \chi^A\left((2\mu_3b^{2} + ((3a - 2)\mu_3 + 4)b + ((a - 2)\mu_3 + (4a + 4)))c + (3a\cdot \mu_3 + 4)b^{2} + ((3a + 1)\mu_3 + 4)b + (-3a + 3)\mu_3 \right) &= i^{ 0 } \\ \chi^A\left(((2a\cdot \mu_3 + (2a + 4))b^{2} + ((a + 4)\mu_3 + (2a + 4))b + 4)c + (a - 3)\mu_3b^{2} + ((a - 1)\mu_3 + (3a + 3))b + (4a + 4)\mu_3 - a + 1 \right) &= i^{ 0 } \\ \chi^A\left((((2a + 2)\mu_3 + (3a + 2))b^{2} + ((a + 4)\mu_3 + (2a - 3))b + (2a\cdot \mu_3 - a + 4))c + (2a - 1)\mu_3b^{2} + (a\cdot \mu_3 + 2)b + (4a - 2)\mu_3 + 4a - 3 \right) &= i^{ 0 } \\ \chi^A\left((((3a + 4)\mu_3 + (3a + 4))b + ((a - 2)\mu_3 + (a - 2)))c + ((a - 1)\mu_3 + (a - 1))b^{2} + (-\mu_3 - 1)b + (4a - 1)\mu_3 + 4a - 1 \right) &= i^{ 0 } \\ \chi^A\left(((2a + 1)\mu_3b^{2} + (2a + 2)\mu_3b)\cdot c + (2a + 4)\mu_3b^{2} + 2a\cdot \mu_3b + 1 \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} + -358602643756079901907475497544a - 134051727355324977339007114728 x^{47} + -458962835374746425167462834440a - 283963371208092953918905532780 x^{46} + (-156859864956300532052110753788a + 484671671714744795153844661232 )x^{45} + (101150545827602970192682512036a + 91243680419765630487454244108 )x^{44} + (587184852342894943490266339708a + 431297191982264154969505495252 )x^{43} + (112482021508537993635220880920a + 501130965042290748239538680080 )x^{42} + (-511542196895706518558335507260a + 166283249599904182449814768760 )x^{41} + (-195092416367996936109698696632a + 332936776716033370195428609816 )x^{40} + (209533928878542887470543532408a + 442397060365683236646007863504 )x^{39} + 149224267377999272176789318636a - 490704883648176704945449548332 x^{38} + 561763358539244813573912869660a - 454823355893785834201720752972 x^{37} + (-336520895822987522129986643916a + 620466137266114152215242665264 )x^{36} + 474477087175394684371866254952a - 210496982888833000293157300400 x^{35} + (404915399920682295826195590248a + 295450509118107677397314166456 )x^{34} + -202179894400150172762583794836a - 603200601539290578318634207912 x^{33} + 171848488449799112869849933448a - 593688800822019346081572759512 x^{32} + (120894909190101304335015050740a + 591224926857031941369185613216 )x^{31} + 232580102911715231704710428564a - 44788553723321197263370314172 x^{30} + 552059680632505524291522537088a - 21437311761800020483216117504 x^{29} + -254525099492422067187894008624a - 274853094538284270360944326280 x^{28} + -118740135543051048191215568952a - 299658003099879897263331885336 x^{27} + 96538229767422869368773160032a - 516053508523855360148431249988 x^{26} + -133414401365890269494008192572a - 155100406645910327110679249152 x^{25} + 325194832773822075990304380042a - 494263904093367477698203029444 x^{24} + (574161780737048107199860980800a + 485730306768641353075830176352 )x^{23} + -130379686389247582893750304780a - 550282815145882423199752595768 x^{22} + (-57687792392600472756362782040a + 121601942457458036506564971176 )x^{21} + (120722930684753571541723219604a + 552524387669086038467090984008 )x^{20} + -400627026002702731880225651604a - 117759345080759405866914691800 x^{19} + 598975974133312597329585795048a - 407186005300724657877749666512 x^{18} + -552330825608567376447696261720a - 417361661285119377833928451048 x^{17} + (190684586701604984230261151176a + 620678615219428492951004040256 )x^{16} + (-90084863588604263247639133800a + 231237178951527139903829716224 )x^{15} + (-534523789299391469753093623788a + 155636531610473915432776845216 )x^{14} + -396540638607962523603282244220a - 385975506245164568244881293008 x^{13} + (247339637786919457759949627944a + 483603886562988399682965814708 )x^{12} + (222468084416076436030782720944a + 422389194208187355115476041296 )x^{11} + 552148225765638311236934603976a - 267071627558439679243866200752 x^{10} + 146474548652906739673648485920a - 38622490195464557845771094840 x^{9} + -156695942632757018118655604304a - 566211157155926585177644382088 x^{8} + (382825583446261753548927709936a + 161854744466321714067077741432 )x^{7} + (-261860760530519154602654511116a + 174774290572087925008801755472 )x^{6} + -385043033941492088118479374816a - 243458563732659555404685325936 x^{5} + (126835196636739557716780319248a + 132753261548154153095528461360 )x^{4} + -222425675288833182734142335272a - 150926420810716321462657362992 x^{3} + (246278133329294127938700635076a + 589442581853832445636656403816 )x^{2} + 610103048360977518171076569744a - 547151965968938235397234681608 x + 191949835867706292834786039700a + 332417422351080243422379611690 \)
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