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ex.24.10.1.33_67_101.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 \(x^{2} - b^{3} x + (a\cdot b - 1)b \)

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_3 - 2)b^{2} + 2\mu_3b + 4)c + ((2a + 4)\mu_3 + (2a + 2))b^{2} + ((2a - 1)\mu_3 + (2a - 1))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 + 3)\mu_3b + ((-2a - 2)\mu_3 + (4a + 2)))c + (3\mu_3 + (3a - 2))b^{2} + (-3\mu_3 + (a - 3))b + (2a + 2)\mu_3 - a + 1 \right) &= i^{ 0 } \\ \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 - 1)\mu_3b + ((2a - 3)\mu_3 + (a + 4)))c + ((2a + 3)\mu_3 + (3a + 4))b^{2} + ((2a - 1)\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 + (3a + 4)))c + ((a + 4)\mu_3 + (3a + 2))b^{2} + (-3\mu_3 - 3)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 + ((3a + 4)\mu_3 + (2a - 2)))c + ((3a + 2)\mu_3 + (2a + 4))b^{2} + ((3a - 3)\mu_3 + 4)b + (-a + 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 + (-a - 2)\mu_3 - 3a - 3 \right) &= i^{ 2 } \\ \chi^A\left((2\mu_3b^{2} + ((3a - 2)\mu_3 + 4)b + ((-3a - 2)\mu_3 + (4a + 4)))c + (3a\cdot \mu_3 + 4)b^{2} + ((3a - 3)\mu_3 + 4)b + (a + 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 + 3)\mu_3 + (3a - 1))b + (4a + 4)\mu_3 + 3a + 1 \right) &= i^{ 0 } \\ \chi^A\left((((2a + 2)\mu_3 + (3a + 2))b^{2} + ((a + 4)\mu_3 + (2a + 1))b + (2a\cdot \mu_3 + (3a + 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 + ((-3a - 2)\mu_3 - 3a - 2))c + ((a - 1)\mu_3 + (a - 1))b^{2} + (3\mu_3 + 3)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} + (-200685740940763987135427008216a + 444980641746562530360638118632 )x^{47} + -559448677681323544843187737480a - 37295093776922201382855053860 x^{46} + (628956280560126820749588968476a + 289561721831491211643383880864 )x^{45} + 327490424354688332367538302140a - 276368464038397048181458601300 x^{44} + -353341845846739681603883131092a - 396617040200432561056584324444 x^{43} + (282368671566164421730122753728a + 89374271680184891315169238896 )x^{42} + -398810631754146921032390044780a - 561424161752409115808995448376 x^{41} + -564725932604807090848863629040a - 184019967382607176861379749472 x^{40} + 470835787095868519438248641320a - 462373067632328590640117290800 x^{39} + (216786269224496215374465212300a + 102070639659799658355271176676 )x^{38} + (162389346654654697595207909460a + 35989515353538615136251264252 )x^{37} + (492625269985952244293595480604a + 42855090722857415703622886160 )x^{36} + (527398104140310471287906351384a + 103096932540381669604310083136 )x^{35} + -375136615226258127299390919648a - 536150072932263837209697149816 x^{34} + -539310394524609801072400831124a - 412547882813755065802343918376 x^{33} + 421684882416922878371879376976a - 415172857301985837293581320248 x^{32} + (-619541849066027172329235084108a + 160323059402657065898037255744 )x^{31} + (244575550598922113516373921316a + 333806791448353241722093757700 )x^{30} + 135255540497886709449534429888a - 475062183864797808302517319184 x^{29} + (-171961430860702916809212186984a + 624532866077626600545624912680 )x^{28} + -484678427202785490703723671240a - 57903793688416541966937208408 x^{27} + (128353586930160141343062641136a + 266032255532422454451978093676 )x^{26} + 54020715873317697678765952108a - 139455337527945640862868065840 x^{25} + (582343885257257308584781441050a + 125871664385749559967392330068 )x^{24} + 10495520506345160306590956992a - 426553861484975774083779769024 x^{23} + (-12148315959912465608477980052a + 52071700598489230402874912504 )x^{22} + 609403308401997795176024142264a - 500289055740092489359693420872 x^{21} + -456136925909505177532270342108a - 175779945636521093437643110776 x^{20} + (-406762831437154733204361503156a + 233835183788951919814530395912 )x^{19} + (-388019165365274606875326741432a + 178590146211336754835635383040 )x^{18} + (-544432845676868243392952551384a + 70526001055644836382745907640 )x^{17} + -353029874852712646964762036656a - 590242769020828641903801571712 x^{16} + (268119874704604558550834872408a + 50923769473483691923634342720 )x^{15} + -274684778200783551981551063212a - 348833354604121299767774488720 x^{14} + -388950087645286871850202587540a - 322299301447325325097014737472 x^{13} + -340879839250093722541815883352a - 606123132545849499005736954044 x^{12} + -51740345917250927837518197488a - 96132931283637997382529119728 x^{11} + (221146353682891819246690608536a + 343526537975629654829148059072 )x^{10} + -472096631228197953179742783312a - 138979984295606871216621323832 x^{9} + (-120714462027508040440875634048a + 63646639378501955343228604072 )x^{8} + 132857215173053593121985975920a - 111398087017183482977088020712 x^{7} + 71014842394709495930521201060a - 398250792442218712878318491296 x^{6} + 86360480157140393960931060336a - 586455999671425239462584841968 x^{5} + (525650067001000877072293636192a + 21611650496240944344849889824 )x^{4} + (582414699429280792154164083832a + 494381035040207601730896152432 )x^{3} + (221249125419775020466792733460a + 170207950179911133485857266936 )x^{2} + (389420824335337339968490981568a + 84201211478102521581488024200 )x + 541718801272607191866931594796a - 259132149148377131251184239182 \)
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