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ex.24.7.1.31_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
\( v(N) = 7\)
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
11
Values on generators of \((\mathcal{O}_K/\mathfrak p^{ 11 })^\times/U_{\mathfrak{p}^{ 11 } }\) :
\(\begin{array}{l} \chi^A\left(\mu_3c + 1 \right) &= i^{ 1 } \\ \chi^A\left(((4\mu_3 + 4)b^{2} - 2a\cdot \mu_3)c + 4\mu_3 + 1 \right) &= i^{ 0 } \\ \chi^A\left((2a\cdot \mu_3 + (2a + 2))b\cdot c + 4\mu_3b^{2} + (4\mu_3 + (2a + 4))b + a\cdot \mu_3 - 3a + 1 \right) &= i^{ 0 } \\ \chi^A\left(((-\mu_3 - 1)b + 2)c + (3a\cdot \mu_3 + 3a)\cdot b + 1 \right) &= i^{ 0 } \\ \chi^A\left(((-\mu_3 - 1)b^{2} - 2b + 4\mu_3)c + (2a\cdot \mu_3 + 2a)\cdot b^{2} + 2a\cdot b + 1 \right) &= i^{ 3 } \\ \chi^A\left((b + 2\mu_3)c + a\cdot b + 1 \right) &= i^{ 2 } \\ \chi^A\left(((2a\cdot \mu_3 + (a + 2))b + ((-3a - 2)\mu_3 + 4))c + (3a + 3)b + (-3a + 3)\mu_3 \right) &= i^{ 0 } \\ \chi^A\left((((3a - 2)\mu_3 + 4)b^{2} + (2a - 2)b - 3a\cdot \mu_3 - a)c + (-\mu_3 + (2a + 4))b^{2} + (-\mu_3 + (2a + 3))b + (3a - 3)\mu_3 - 2a + 3 \right) &= i^{ 2 } \\ \chi^A\left((2b^{2} - 2\mu_3b - 2a\cdot \mu_3 - 2a)\cdot c + 3b^{2} + (2a + 1)\mu_3b + (2a + 1)\mu_3 + 2a + 1 \right) &= i^{ 0 } \\ \chi^A\left((((2a + 4)\mu_3 + 4)b^{2} + (2a + 4)b + 4a\cdot \mu_3)c + (2a + 2)\mu_3b^{2} + (\mu_3 + 4)b + (2a - 2)\mu_3 + 2a - 1 \right) &= i^{ 2 } \\ \chi^A\left((((a + 4)\mu_3 + (3a + 2))b^{2} + (-2\mu_3 + (3a + 2))b - 2\mu_3 + 4a)\cdot c + (2\mu_3 - 2)b^{2} + \mu_3b - 2\mu_3 + 2a + 3 \right) &= i^{ 2 } \\ \chi^A\left((\mu_3b^{2} + (2\mu_3 + 2)b + 4)c + 2a\cdot \mu_3b^{2} + (2a\cdot \mu_3 + 2a)\cdot b + 1 \right) &= i^{ 0 } \\ \chi^A\left(((2a\cdot \mu_3 + 2a)\cdot b^{2} + 2\mu_3)c + (4\mu_3 + 4)b^{2} + 4a\cdot \mu_3 + 1 \right) &= i^{ 0 } \\ \chi^A\left((-\mu_3 + 3)c + -\mu_3 + 3 \right) &= i^{ 0 } \end{array} \)

Inertia Polynomial

The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + (484144114902291511791809789696a + 191615613701189410339093005440 )x^{47} + (-73708923751823963766131503824a + 196787096447730417787442622384 )x^{46} + -76056065061064013319976771384a - 168989410843244789702865255776 x^{45} + 457544232535315942767627095704a - 185728205148439250868383212712 x^{44} + (-114885063557587535588497606008a + 255305573841594849189823210696 )x^{43} + -472791500302547985875394172036a - 611326606656198391353847218916 x^{42} + (-258495196978321418781832221904a + 174547153334880874223013608944 )x^{41} + (-25751067900875140356758149448a + 78229331357952171379691829948 )x^{40} + (409344063110001384295826220504a + 494298225600540362971947112704 )x^{39} + (-380561269296169330453180384104a + 356100409069551319735631036736 )x^{38} + (-79602634380049478108237713672a + 298723688208880341241260993968 )x^{37} + 70516499909364838522780623364a - 537673290132057842598719857244 x^{36} + -284147322310804371984446595936a - 69489530188163043042854446432 x^{35} + 142604215942451339032774511552a - 323439269225161158537526451440 x^{34} + (218725849603448847378257239944a + 21231046325337783499628190064 )x^{33} + (449670094383762773265465390924a + 234965434304439415973441588224 )x^{32} + 393007062553393712784849310304a - 527117489534629999710149201568 x^{31} + (44334058551207555630143336824a + 514150467556550728640136293720 )x^{30} + (224514657939586328380246201176a + 126194941570423316349483893872 )x^{29} + (-624887522361142275582788937412a + 328849991849164948160343691556 )x^{28} + -508378485266505786495634987472a - 423863376591226248258150413432 x^{27} + (407308625453173436469838187920a + 166015412230972434018628391584 )x^{26} + -106191059138688677477682757736a - 73014848532951934592720943824 x^{25} + (493349455550314640287672156214a + 498563882456935446052054020536 )x^{24} + (-157154309363359592376195237920a + 264949635014970933568360978016 )x^{23} + (-506529176022865760127710173112a + 312216753057111165264258138432 )x^{22} + -98880185533500076888492476672a - 434222560799381408603015169808 x^{21} + -596980663418592916585192262488a - 324151141981482029842121660528 x^{20} + (-93351123313879297112854843000a + 486985769467874323200357412096 )x^{19} + (-626584600846188272299255871880a + 498255140840609081349425783672 )x^{18} + 413293208776235312677932043520a - 458826566262305578973498109088 x^{17} + (321176828143647288976205880852a + 533317071221023993574072093856 )x^{16} + -246078390315028089576172548960a - 156400032626039624950031894736 x^{15} + (246186348011365433600413312984a + 558348435852689376343565273776 )x^{14} + 476074372395790657404150789024a - 567881718673108974164993486912 x^{13} + 564504496186340841233371680040a - 465871577394825590824063531100 x^{12} + (-620954996077836115069609587792a + 357682570662842696726900785920 )x^{11} + (-16641685594754437298613574512a + 382711069623182244231630461896 )x^{10} + (152296773240572906728544189400a + 320802730484962318226383890480 )x^{9} + -577686343997060187239764569048a - 54719161902842196698173579352 x^{8} + (-292443422127872206165731191264a + 455203428529985640006347342144 )x^{7} + -287646510706612829736102764904a - 168929047872041265088911794624 x^{6} + -25906453960309151945668441552a - 547033160806571732419260754592 x^{5} + (-551746911234693100693399500076a + 94379966023278312941561327440 )x^{4} + -183380454586207024902735507440a - 315654617116644909251902834960 x^{3} + (-379144992968237631529430253008a + 320946479196878069311405742464 )x^{2} + -506919686669972162894828036120a - 38977726472895234261980419296 x - 255244656388785827715914192436a - 323185916104342104916841520914 \)
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