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ex.24.10.1.33_67_101.d

Base Field
\(F = \) 2.1.2.2a1.1 \( = \mathbb{Q}_{ 2 }(a) = \mathbb{Q}_{ 2 }[x] / (x^{2} + 2 x + 2 )\) 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 \(x^{2} + ((a + 1)b^{3} )x + ((a + 2)b - 1)b \)

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^{ 2 } \\ \chi^A\left((2a\cdot \mu_3b^{2} + ((2a - 2)\mu_3 + (2a + 4))b + 4a\cdot \mu_3)c + 4\mu_3b^{2} + ((2a + 2)\mu_3 - 2)b - 3a\cdot \mu_3 + a + 1 \right) &= i^{ 0 } \\ \chi^A\left(((3a\cdot \mu_3 + (2a - 2))b^{2} + ((2a + 4)\mu_3 + 4)b - 2a\cdot \mu_3)c + (-2\mu_3 + 2a)\cdot b^{2} + (2\mu_3 + 2)b + (-3a + 4)\mu_3 + a + 1 \right) &= i^{ 2 } \\ \chi^A\left((((2a + 4)\mu_3 + 1)b^{2} + ((2a + 4)\mu_3 + a)b + (4\mu_3 + (4a + 3)))c + ((2a + 4)\mu_3 + (2a + 1))b^{2} + (4\mu_3 + (3a + 4))b + -2a + 3 \right) &= i^{ 0 } \\ \chi^A\left(((2a + 1)b^{2} + 2b)\cdot c + 2a\cdot b^{2} + 2a\cdot b + 1 \right) &= i^{ 0 } \\ \chi^A\left((((2a + 4)\mu_3 + (3a - 2))b^{2} + (2a - 1)b + ((-2a + 4)\mu_3 - a + 2))c + ((2a - 1)\mu_3 + (2a + 1))b^{2} + ((2a - 2)\mu_3 + 3a)\cdot b + 3\mu_3 + 4a - 3 \right) &= i^{ 2 } \\ \chi^A\left(((2a + 2)\mu_3b^{2} + (3a\cdot \mu_3 + (2a + 4))b + 3a\cdot \mu_3)c + ((3a + 2)\mu_3 + 4)b^{2} + (-3\mu_3 + 4)b + (-2a + 1)\mu_3 \right) &= i^{ 0 } \\ \chi^A\left(((2\mu_3 + (2a + 4))b^{2} + ((2a + 2)\mu_3 + (a - 2))b + (2a\cdot \mu_3 - a + 2))c + ((3a + 4)\mu_3 + (3a + 2))b^{2} + ((2a + 1)\mu_3 + (2a - 3))b + (4a + 2)\mu_3 + 1 \right) &= i^{ 0 } \\ \chi^A\left((((a + 4)\mu_3 + (a + 4))b^{2} + (2\mu_3 + 2)b + ((-3a - 2)\mu_3 - 3a - 2))c + (-3\mu_3 - 3)b^{2} + ((a + 3)\mu_3 + (a + 3))b + (a + 1)\mu_3 + a + 1 \right) &= i^{ 0 } \\ \chi^A\left((((2a + 4)\mu_3 + (2a + 4))b^{2} + (2a\cdot \mu_3 + 2a)\cdot b + (4a\cdot \mu_3 + (4a + 4)))c + ((3a - 2)\mu_3 + (3a + 4))b^{2} + ((2a + 3)\mu_3 - 1)b + (-2a + 2)\mu_3 + 2a + 3 \right) &= i^{ 2 } \\ \chi^A\left(((2a\cdot \mu_3 - 2)b^{2} + (3a\cdot \mu_3 + 2a)\cdot b + ((2a + 4)\mu_3 + (2a + 4)))c + ((a + 3)\mu_3 + (2a + 2))b^{2} + ((a - 1)\mu_3 + (3a - 1))b - 2a\cdot \mu_3 + a + 1 \right) &= i^{ 2 } \\ \chi^A\left(((3a - 2)b^{2} + b + (2a + 2))c + (2a + 2)b^{2} + (3a - 2)b + 1 \right) &= i^{ 0 } \\ \chi^A\left((((3a + 2)\mu_3 + (3a - 2))b^{2} + ((a + 2)\mu_3 + (3a - 2))b + ((-a - 2)\mu_3 - a + 2))c + ((2a - 3)\mu_3 + (2a + 1))b^{2} + ((3a + 3)\mu_3 + (3a - 1))b + (-3a + 1)\mu_3 + a + 1 \right) &= i^{ 0 } \\ \chi^A\left(((2a + 1)\mu_3b^{2} + 2\mu_3b)\cdot c + 2a\cdot \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} + 118316769627035975998551516572a - 238598229279484462010323883136 x^{47} + -205087786788804212840747287340a - 508214468919547935799220986700 x^{46} + 531101820970482208297997161808a - 26715840163865890034399498200 x^{45} + 558558662831996781496911387876a - 497149303575493704811653657228 x^{44} + (5899707959172335111945927900a + 186423180357544632582197678980 )x^{43} + -64604957603224382745897798764a - 552941472608140159489298809976 x^{42} + (-224131836151406142036418728580a + 179433770057589142035065552728 )x^{41} + (-74847921032540259423437668380a + 517910183175755093091378544568 )x^{40} + -211010335991355702506161656488a - 317335226396777236300766038040 x^{39} + 177714434982329773237091063644a - 240994615458847536942501441236 x^{38} + (234426839021768806758097639224a + 632359994803039111711876259636 )x^{37} + (257558918235999211709060719648a + 383746118247317259307884564912 )x^{36} + 516272361688808951560016073776a - 182789206897763321458028465512 x^{35} + (287236740342234617632528920128a + 327343600014842683458187326952 )x^{34} + 534903521134303377867459252284a - 501216593671505969992250065608 x^{33} + -414915054189256772145183156276a - 79766485324428665873851785632 x^{32} + 365785451561054806823959554392a - 301515696914846685127322946864 x^{31} + (14357441591286794025226134072a + 178292719611781531136367645196 )x^{30} + 78304817128974768499128306432a - 552144522641932072453435607152 x^{29} + 445354983713084864594825834832a - 595406306551412137259798659176 x^{28} + (-630109240735280587977923580600a + 398572780473405126615380374216 )x^{27} + -283517656374896589222983011728a - 342310487853295931016293546924 x^{26} + (-528971554778661373716483405272a + 532287022506840512736537017624 )x^{25} + (569438596525405826367323220902a + 317100741382408875681801750740 )x^{24} + 257307777007745203757605016200a - 108313375120608507974217195192 x^{23} + -124675295486689386814074061580a - 217193507249009563315571899568 x^{22} + 33831968334426346084363361000a - 606825906208850013798151350816 x^{21} + (-513434196444100820603666037332a + 567980918468556193657129740024 )x^{20} + (388823112493404476539893666332a + 183888155329819375600092552248 )x^{19} + -105157785326984403383901562328a - 501503895660503117218278564024 x^{18} + (29797547944808158468341130800a + 363596595119751560396107228648 )x^{17} + (-259796138176143514388038249928a + 379271099906279982418603451176 )x^{16} + -427527241103726313243612917504a - 82522473173088865907672216064 x^{15} + -264178564163228141657563009556a - 276679408977376639073068634576 x^{14} + 562111527662529766526330348236a - 461393721463254731387831092856 x^{13} + 287261956750507570522037124432a - 360404884597943308328249418252 x^{12} + -46095152307999629040751390744a - 581139164972129657345822017216 x^{11} + -247271169096796437882344712616a - 57161922001208675045418003920 x^{10} + 264551406208712559116012220376a - 441302361256431098286598132904 x^{9} + (-612254939656773591047068862288a + 332614307753591211566563781552 )x^{8} + (246970870933560025315975383872a + 45555400366093327114386108432 )x^{7} + (12656459434990218849294755404a + 331537548814620762335677038840 )x^{6} + -176620812810858778018947151792a - 301107053513172277347082404816 x^{5} + (162040628471977604773648779888a + 561860331071688340751752863472 )x^{4} + -382585688574229053028738251864a - 198898469442439827447437186496 x^{3} + 44974960217652236941519929060a - 3610531251607502075038190632 x^{2} + -555625910225795302557563232720a - 50469121398454298281969274112 x + 200391450062249669914074118890a + 527268688373079247661149323874 \)
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