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ex.24.10.1.31_63_95.d

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

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} + -554290560199530622421154316828a - 41493067463679530603576698504 x^{46} + 509971061977007502432924498144a - 392577418429029356200609359400 x^{45} + 229265351983821853401092453132a - 326213863604756144923802425592 x^{44} + (-385422564546654316202489105064a + 223783793319493840593238517436 )x^{43} + (-465571537034271684149772264576a + 438806839111680867444152903672 )x^{42} + (-613926324125551213701110075272a + 99152811498180698981454665976 )x^{41} + 205343979920596145388824977448a - 516897342227995562501498954880 x^{40} + 363584264054314921296715786824a - 573403380730411665149397749520 x^{39} + 349963745353420698328643666528a - 30715903935289071011118102092 x^{38} + 619791001576035400135872706268a - 601682337095635461640635850528 x^{37} + 565838307824919751640567721488a - 549663016852760749751122488584 x^{36} + (349830557183530029794133449640a + 274342212010120370174694673648 )x^{35} + (454229244478456815459149537664a + 582392654537485995813034917104 )x^{34} + (481445241475973336004829802744a + 600921181290241790463563670872 )x^{33} + -574372767267717655386519096640a - 289912286829587410492225693592 x^{32} + 310467597313661621661542354832a - 347162215224404771441993341472 x^{31} + 207520731327446886237832264800a - 153895539464111115615086094500 x^{30} + -51050810284117877161568736876a - 516789956413296398717833570288 x^{29} + (-428413429271246573847283029360a + 249506791455706073597233659640 )x^{28} + (-105839928586620377973600985472a + 34998094909642889559710452624 )x^{27} + 212322675030488984058677057224a - 534973443436320484666000303984 x^{26} + -54420222262018255094832094720a - 73972110703144277819831104256 x^{25} + -114920961263168621052733257474a - 38080060216395800865453710184 x^{24} + -522944429726933734290346494912a - 613952246099382863065126119400 x^{23} + (84807423354949481124458476376a + 171742044378956023481827264712 )x^{22} + (-513564618844629986835632849016a + 31598222134798674042581084352 )x^{21} + (453325625849499414431105027392a + 630356063320336592226505448064 )x^{20} + 432826849432401942204041553500a - 107620404433265796856446921792 x^{19} + -531433212321684238764559344536a - 329445450510002626128529481248 x^{18} + (283126201864914276623753353576a + 291566155887335713927150658464 )x^{17} + (66642643724499697500808678768a + 375964552664681377344734719552 )x^{16} + (-627729761652734112158111667840a + 223578601105212611962318344768 )x^{15} + -45917213811335009047902374260a - 562652434173135106908127868448 x^{14} + -45128368474960493377854752784a - 130387658297303939241133265944 x^{13} + (45656519130147330269933684672a + 314410385504055878093730626468 )x^{12} + (-319975474267343641289573439256a + 449052578532699964967690426896 )x^{11} + (-74144752260940530581541679680a + 342622294193804642016611163200 )x^{10} + (389293965795223935319571004616a + 155959582548169980811620116336 )x^{9} + (135662783661783107855251794968a + 251161425778967336903990919600 )x^{8} + -28064932082489442899654624240a - 152963811159779773731176382096 x^{7} + -323548710691373994895033011764a - 8056676473644605517645334672 x^{6} + -320945506279730120827506964064a - 554934845069630797398447443544 x^{5} + (-516737581657670020319866068360a + 355955892968271371479383369392 )x^{4} + -610797020367841944915701522400a - 412225556523643957884286643520 x^{3} + -396520636460331870724200927952a - 29905062717653077699443209120 x^{2} + (389109977318191160985315058968a + 564437243764329270394355911168 )x - 394107352057892174554125508872a + 535004093303230583963483621134 \)
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