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ex.24.10.1.33_67_101.c

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^{ 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 + 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^{ 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 + 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} + (-30889817246056100882105349152a + 258054856792555961737664664236 )x^{46} + (96904121339673572997014163004a + 320616633256344600207141659776 )x^{45} + 541303725913829698530833010684a - 260805876452182338716900099988 x^{44} + (-220776073647768069350777771684a + 292240963424960885560347995348 )x^{43} + 377041752474260944326067393472a - 518284025704164207226457892624 x^{42} + (306055248714319074136167732140a + 244678155200581068289040299768 )x^{41} + 358597100534951642986774281816a - 616842909751226589211833060544 x^{40} + (156341347499883207113882099096a + 184316902828069064134394964304 )x^{39} + -66464018885117526612419076772a - 140381220659730089121051620412 x^{38} + (456882497882673944968696693260a + 234967607182251841047302673964 )x^{37} + (380991660734684903031194473404a + 50892155361246179680930882624 )x^{36} + -487042275607880070654799319848a - 567268102799984878600251989424 x^{35} + 586097763977222474890981126496a - 32200607991372585744138963208 x^{34} + (-494798329246115803445367370020a + 84131497096703515817542271512 )x^{33} + (-100790335214080884272666202312a + 419467807073279248205568855480 )x^{32} + -305159149177717247499384750668a - 105833085777383924001361851296 x^{31} + (-531580549423556188436776395820a + 326815538579583402134499629108 )x^{30} + -618801612883395851527447388120a - 187337190532688724200080735696 x^{29} + (-213131980656204268183839688152a + 41102237518124908260397606776 )x^{28} + 290586540389474722757963071160a - 334934378776817551402731490296 x^{27} + 401535837081305263479942835184a - 66656393340500754638519380820 x^{26} + -79554513570604862546412937508a - 526846710994670171375296727664 x^{25} + (3896067012841569538121395642a + 244926775256166399078951792132 )x^{24} + (-92722138201797033702656459488a + 488141905304054694922010470944 )x^{23} + (-169907125851662604633618309796a + 350121048249308347506498543016 )x^{22} + (-82488172397166580844346206824a + 368028214278528132656995294872 )x^{21} + -577925260926416414860219078492a - 242845730821891454967487809048 x^{20} + (-343232738653540875374861002836a + 582367037920222196103475291048 )x^{19} + -351293012128263248812548972600a - 405234566016406122798538136544 x^{18} + -199205213768877536994045307448a - 217552052611304271663757652024 x^{17} + -328545483070503648253371941312a - 419517079437878774681742403632 x^{16} + (195861736069682236710181994872a + 394509821998697540217438067616 )x^{15} + -607093284466557034191735867612a - 475577466067615834851580887184 x^{14} + (-520306942016557240551614151700a + 71955405497106428700129511280 )x^{13} + (-353908986999858485943655839720a + 536795490430128051387435388868 )x^{12} + -275439623628736328479717068368a - 236899049363249775628405685712 x^{11} + 294394386860709787622673070696a - 58771360816241275982766360704 x^{10} + (-2277366563337264399447481616a + 144714976902072021848547805832 )x^{9} + (128885399519279174116229278304a + 373132736342497220750171951640 )x^{8} + (541581863349378335515572638672a + 269188424756973750251972834808 )x^{7} + (-457616129132913404102005497964a + 259916202164450214068149225440 )x^{6} + (304017406749077929306418073888a + 123337806839576287191325681088 )x^{5} + 577554138514081834295447827440a - 484132862669349711012242386304 x^{4} + (191927873922201835492047920888a + 443765641525529073261939813872 )x^{3} + 201474926597141907888175818804a - 470734538521993914194359333352 x^{2} + (-446088450669973511730691758464a + 420338830959919270293636797192 )x + 78295308963199837979932285500a + 482026081047950980418234767050 \)
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