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ex.24.10.1.31_63_95.d

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

Inertia Polynomial

The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + -97852459047642662183674462436a - 628524657377182367834157856480 x^{47} + (-463021037477310337491960317180a + 494234605094738290028321796688 )x^{46} + -471245363322414097409057330824a - 34400919853827512739265694008 x^{45} + 90860029947111490193477672196a - 276867741458250619586853409008 x^{44} + 534294931315786309776927927088a - 127263967339286641952054286980 x^{43} + 504416687566984167579574453028a - 187422703878780143804837191512 x^{42} + 441394060676227810376767125664a - 211476911007110182562783676264 x^{41} + 442483407178640892267475931448a - 473135166472021183100115754040 x^{40} + -491101233977578073341385356512a - 152374522797923731366553644400 x^{39} + -344108820363196142278160047224a - 75790332692394761722491869452 x^{38} + -249721299216268885235058010840a - 142637848503326127573195560288 x^{37} + 479173751136585127916243859788a - 19491633511586372545128430560 x^{36} + -163401920261506629749515500208a - 338500798368796799722140294088 x^{35} + -614462701229458808213741977244a - 547988112786478987958260221520 x^{34} + -263162305766089184323436887544a - 21985138425609365801415662040 x^{33} + (529753282449480173784665562476a + 474654730901304481975964599384 )x^{32} + (279912564812326891616721241508a + 363415386760662372830531887024 )x^{31} + -544136226621186715829274885304a - 598113009643890797881825403596 x^{30} + (411552191247349003112145306624a + 357018929951873912579633523328 )x^{29} + (-503416708842241520142835545384a + 192003087094445186533621159184 )x^{28} + (-356760167724368455335825023120a + 150385207446745373292587057440 )x^{27} + -220201955944885151680456215088a - 601530894344114386847009723560 x^{26} + (474856391376375023925570679832a + 424843233029508077158558696480 )x^{25} + -577548289893098256985991984978a - 342444155439112499326524975576 x^{24} + 190062001591781364599875410960a - 171724646837393971017268024904 x^{23} + (-367845001521864177223530173936a + 511071107370289661742578618456 )x^{22} + -460879257304426359000440044920a - 487039986150977814508903534384 x^{21} + (245431653542671583168318255704a + 383516728013428423961056216016 )x^{20} + 308227594455509745540232604388a - 642803220132233817456967216 x^{19} + 85099443035825166285419833616a - 103456698381905478147031146488 x^{18} + -399195882068425347401660982824a - 280652363823016693000101157648 x^{17} + 46135149530001916387613675920a - 439156801355348908885637705312 x^{16} + -35698463561360656221169868880a - 495737909208187309850384052784 x^{15} + (-267944675952139803943916630204a + 600577946365099892697944793312 )x^{14} + (-534373944055641155995637673432a + 363784137044741259523705708016 )x^{13} + -167381409395441697222399804368a - 223656368322465617652953776660 x^{12} + (-81431651954833794186261364584a + 179703701139325956339307365056 )x^{11} + (-40582696344147509438258094384a + 342654538616000405848268417096 )x^{10} + 530276928529720748808393915160a - 355084681374479928624417862576 x^{9} + (-117585619611435211065460494856a + 552276950693466105560535763672 )x^{8} + 361215138668210816399633685328a - 128138803050334539219260583688 x^{7} + (-33018791992359447706724471948a + 570779250863835320019334143040 )x^{6} + -494556865644002420468628446960a - 301185710274602585198879435936 x^{5} + (-102192565514467796564774276144a + 138745589511745530816364403520 )x^{4} + -250535701488170275559424103456a - 486499248792826919338710527968 x^{3} + 56303836521119194988374155464a - 508152052513834468162609118400 x^{2} + (631531526838776744114716651000a + 266895534403948647170571949488 )x + 543347539471510484663414576992a - 38839258948352074364061187230 \)
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