← Back to 2.1.2.3a1.3

ex.24.10.1.31_63_95.b

Base Field
\(F = \) 2.1.2.3a1.3 \( = \mathbb{Q}_{ 2 }(a) = \mathbb{Q}_{ 2 }[x] / (x^{2} + 7\cdot 2 )\) 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 \(x^{2} + 90546471444873528678335943241b^{3} x + (90546471444873528678335943241\mu_3 + 90546471444873528678335943241)b \)

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

Inertia Polynomial

The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + 28261683353735459005718532588a - 6821935044214331538437044640 x^{47} + 122253372085648713198594290500a - 606220493908471065292451962448 x^{46} + 131408971380803544314176513584a - 136642601081315899853358479328 x^{45} + -572806060099403811302995989664a - 602827936386525094462758925600 x^{44} + (531186288938066898599523114120a + 624137601198981663964399757860 )x^{43} + (523867334449902298326623867500a + 41217238116696714389311718984 )x^{42} + (599649631903143568350877491040a + 125890936724669082913672625496 )x^{41} + 623374510612923455990698034080a - 599717570882680770658776665736 x^{40} + 196074775417083940318341757952a - 191959996758572108152897268624 x^{39} + 81609706219233026459932118992a - 493491803842228464671124657228 x^{38} + 115109089097693084873416375048a - 235904985476389728120790508840 x^{37} + 402604158587608298395743086080a - 73181378218699986714084139624 x^{36} + (-193901442507601236496712215264a + 20606497297168075872776143224 )x^{35} + (-373798216702649228680779438940a + 429982217029310608306426688896 )x^{34} + -492911800489687675918587098920a - 481230901450665629322163662792 x^{33} + 324550112943347233975307070836a - 210233394959392023138673147064 x^{32} + (-480457555884939044794260768688a + 210562264684709268844496997152 )x^{31} + (587777682511022813323871667200a + 145000960358995453162301743492 )x^{30} + -165300748722393519099890273288a - 57409396631931177735696179712 x^{29} + (-57822002349670608661977585968a + 484531738597238309179895424384 )x^{28} + (-19169783259576529957501119792a + 63941250867001393162544309584 )x^{27} + -144475423072030719417295743872a - 573522558742170137846935111016 x^{26} + (146140158315120636126848687664a + 46667205885405420400697782872 )x^{25} + 56792014798548364016946164638a - 441331811571906017145860333112 x^{24} + 445386400405065246345867205936a - 206559829744599953305542013416 x^{23} + (493537852208345871602215877040a + 164361951030101704029952594312 )x^{22} + 96227494810871951635195771040a - 204605312199310491799579572480 x^{21} + (-199035012665031273431891947064a + 622569194303362791561596973320 )x^{20} + 23506674361904443465207773340a - 387849658661088337025776442640 x^{19} + 510289741830104023215836843144a - 182159520027726858310937753720 x^{18} + 239029838000088129375870439256a - 139188057115416434314187356928 x^{17} + (-15488301265239218002140850272a + 525791570316109252250324273200 )x^{16} + -197782296881720876124638496640a - 310363265278890683160449819376 x^{15} + (573295115425326427440474026804a + 332687499783722626232145801408 )x^{14} + -512598050765837872552774902792a - 234484192228295820987179863792 x^{13} + (234694841450890817444060950680a + 123372542546271142788335540388 )x^{12} + -63112714114710669113515434504a - 536302661604810802079229676704 x^{11} + -93732165271118597004675196064a - 251728645658425819553480326072 x^{10} + (549876769201523582660714571336a + 467105973693290763377791766960 )x^{9} + (439918647236616192942229591736a + 52516681282553531405836993256 )x^{8} + (79932500227894702526162695648a + 12991558809738784351875548768 )x^{7} + (-364477460178060122960619074092a + 579413422097489895368684591504 )x^{6} + 35600682480350480618433722752a - 490003507334569254742509249904 x^{5} + (-415501447948799528487879877184a + 27737732743531576178054686160 )x^{4} + 161035974031242839973052328896a - 31257642217998740612521948608 x^{3} + (-390146299426902045700465751400a + 577248845478869998818843279200 )x^{2} + (-200806300952817429741835141744a + 610092902024762322778311810496 )x + 303511184502898565236030715240a + 471387211656535233666705212114 \)
← Back to 2.1.2.3a1.3 Summary