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ex.24.10.1.33_67_101.b

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^{ 0 } \\ \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^{ 2 } \\ \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^{ 2 } \\ \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^{ 0 } \\ \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^{ 2 } \\ \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} + 492224828540059726358751356140a - 195848183831026027024308083544 x^{47} + 67170049868766717298753161588a - 268365276958488025245442152116 x^{46} + 207561894041291346310143431480a - 411372239497813028553430423712 x^{45} + -232231573327565132572461349736a - 49232493378719400410797774508 x^{44} + 514908359051941811412111136620a - 361491986799456988301543567268 x^{43} + 587167001144040595597853149080a - 451700214476976833476386052232 x^{42} + (261912455084263593109971440708a + 386103100226636303303371743912 )x^{41} + 623557874685166271568379722676a - 4005510308333061456505656768 x^{40} + (-607410874923796486813941693024a + 157378523546399303450452623064 )x^{39} + 57345193284863549918222234428a - 464905074748103681459765513780 x^{38} + 602034541141101765173047695932a - 176819730783628497907694081836 x^{37} + 230512378427182614901517812616a - 240559446796563469784826042664 x^{36} + -28449576224924755172172438752a - 187569795209719519625692830664 x^{35} + (190707284053794459754025524152a + 21294526226589498790706449200 )x^{34} + -546633853893439356212970201964a - 43633948542209586621904275944 x^{33} + 167732438920129412099341641244a - 366607407805301985083928057216 x^{32} + (-142486652997859914326692155032a + 631615574631517198742753556576 )x^{31} + -471286000506686124320361357596a - 537525980476110313583500306076 x^{30} + 467279153870035693580667848888a - 573075038106118781938908929368 x^{29} + -611522664713954699674814307080a - 307271558845057872769161780920 x^{28} + (-486975613951970277461598187208a + 260172646674226212361403955544 )x^{27} + -138925417594670388897934616920a - 407065869129930546287808235716 x^{26} + (387756923793717008704569678176a + 372617049569379750916253697408 )x^{25} + -463508204050144529894291197862a - 357713900429744438441943809204 x^{24} + (-175371230506889174366700118392a + 520197325196755170489891437560 )x^{23} + (-87205293127485782548471948564a + 427157113256359754132273135632 )x^{22} + (-591648401003746587726412843552a + 454367636762362892390842224480 )x^{21} + -539336606469181254398386574124a - 320319783793215053598985498880 x^{20} + 382331825337066268207274298932a - 443215270833897240476498103320 x^{19} + (500630275189752898412270744872a + 20693275511734928900346078336 )x^{18} + (-523293129666965783830388953784a + 457415406348015736373467797448 )x^{17} + (622686752761199621064461614472a + 595509723228002110665153729736 )x^{16} + 98913804744049769331155507808a - 44821857751701691493314566336 x^{15} + (252908615912045078065725667212a + 342546750445737776132521061264 )x^{14} + (-47757361856332805283428368740a + 265891820189219104074263583456 )x^{13} + 274778279205489712593496582248a - 569893938150683411104665756580 x^{12} + 77427782434277160390646349784a - 474573384649103967100575517216 x^{11} + -567427513565616254171452714416a - 155141440102804783038469442608 x^{10} + (-627704636120628748565369324544a + 619880573311610461803681445816 )x^{9} + -422631659412380858285296511824a - 269433341497273459324170894480 x^{8} + (-272566082883791407212770015120a + 446301999853509121996113286640 )x^{7} + (514300253526929715218943444564a + 220467432391496626397986691152 )x^{6} + (88917049956806918283252558968a + 495162613033566906920688106704 )x^{5} + -81746706754417787618286076144a - 578914740283721473742760862032 x^{4} + (-93550977367303444594756777960a + 538331368699917189325952816384 )x^{3} + (-497120035268774532673211574684a + 530593250952782216965560105864 )x^{2} + (-534971507040490506820671319416a + 262778772936066607672572450288 )x + 38574777151807907674312723004a - 509426689225259873199132430030 \)
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