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ex.24.10.1.33_67_101.b

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^{ 0 } \\ \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^{ 0 } \\ \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} + 439786656758452492784345844464a - 196313674962279585856608394904 x^{47} + -259573939307391293965381400264a - 605372013979619401064498319476 x^{46} + (175679013941196901590991891860a + 506812710183773243924121752216 )x^{45} + -220318659482239918643812662576a - 259495641378546665799233903036 x^{44} + 504073963561997587724865145116a - 375060559270085160996301642940 x^{43} + (-632052041351416318224328816220a + 518048074576761225835600137120 )x^{42} + (-225989302049387688511237983940a + 410178745735570090249784002376 )x^{41} + (587064114782377141431187798808a + 89289045688567454667426198936 )x^{40} + -585499557467880452050421916656a - 200701094758901018606333164976 x^{39} + (488239828921570598506781040804a + 320393836775984921112951261476 )x^{38} + 263546436876205042752199531592a - 462030433564657191935287839556 x^{37} + 613672864977027932061435013868a - 285419049332829353858106238184 x^{36} + (432797325568093844418030521192a + 201293537480431568178126628720 )x^{35} + (169265477176483162971664434480a + 49771262648852234566858304256 )x^{34} + 412295338352612420964896074132a - 422325185629780729906575179080 x^{33} + (270849123144586423726536550488a + 577599478920110125003860284496 )x^{32} + -38283289821028631115416453644a - 395992447788206197647798549456 x^{31} + (499537494024784511649741142952a + 425388117610051458302162646764 )x^{30} + (-312577641100373849700262969152a + 249772941617357304997604383464 )x^{29} + 39991770280286731771148425384a - 491418380586575923216777235064 x^{28} + (-115203281331066726413202735784a + 64557464046903055439183350056 )x^{27} + -465775157099467237863304305416a - 91254922304309196873580761052 x^{26} + 607797626902066917355356057428a - 431870176194299936224841738496 x^{25} + (-31537222367972379247360357210a + 426295533982267064207495368236 )x^{24} + 474634431089860926675392135056a - 550416622959799100354399514032 x^{23} + (-325479003620169943138890260332a + 133032678586199094968527623176 )x^{22} + (-611329540313245322680466175192a + 277615451331677220045386363176 )x^{21} + -271633395821285438310500887588a - 491517295307968782820653158288 x^{20} + -327081048467792375944377970900a - 327095169851150582006227430824 x^{19} + (-84332515885950727928455872472a + 515440313170308870580270747112 )x^{18} + -555426450371205667815555056000a - 575184860113455476391610148584 x^{17} + (109420280910015895167972953976a + 198918480158645115827413773312 )x^{16} + 203717683970772232609727385944a - 382077408402374221024135656480 x^{15} + (-250211166795057040276266899116a + 96718432489448992740399699008 )x^{14} + -87636734984924985093365556596a - 148304024671031135241001254200 x^{13} + (402591260417574684270865099752a + 108355239060142701332299955292 )x^{12} + -115830012344761056048923917072a - 120788636910376523501746937200 x^{11} + (-566173944411845184361374697392a + 436975545371078667996804714384 )x^{10} + (477332368180323543269437096344a + 408278050844239678346782758344 )x^{9} + (-311007342985723501386776485664a + 485159994993738284941071919480 )x^{8} + -81729818145693168865862412216a - 219903489014919507314786328248 x^{7} + 584191830660628054057746290940a - 454146499029582619845281163128 x^{6} + (467347874017663833095314440472a + 34432293835349855311866684448 )x^{5} + (334405799260037516878236135264a + 329857447706169317157853620752 )x^{4} + -512369315814887736557926839720a - 166898850713434016114690453200 x^{3} + (-579077843797969326640232782076a + 340959961109214017445814704840 )x^{2} + (328903909691064288425657499160a + 311752437051598683382468557048 )x - 620031803327477525335484540342a + 196228392005135341352204646578 \)
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