← Back to 2.1.2.3a1.1

ex.24.10.1.33_67_101.d

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^{ 2 } \\ \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^{ 0 } \\ \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} + (605441474450544305441971075588a + 439907583548591994110995147908 )x^{46} + (244529394502099606045032282752a + 394320546198903679090963776496 )x^{45} + -543325511051928910928476790128a - 372905780126784851307162103532 x^{44} + (-2374935173303082590242771076a + 55427100275214679810311494620 )x^{43} + (-99458747233527121054494446160a + 513683876801810027563546216120 )x^{42} + 321615362023392831770371881852a - 207526801271773567832567084536 x^{41} + (-252748754312628875033454789580a + 368768713937528563353750588376 )x^{40} + (-186488312755214789589131995456a + 291861171511076614417528629016 )x^{39} + 562777511005666847284361769484a - 51002318633216410029865803396 x^{38} + -6346719295636646759289796908a - 104099273233556702411409444900 x^{37} + -488133884957241554324170634320a - 198639936935094642892658603720 x^{36} + 445769151983628026210798573664a - 622793370158536387988246284936 x^{35} + (-406670739436237175748814378784a + 215291995338335139083337871872 )x^{34} + (238225131490683326006470514676a + 558331884895948798665651869224 )x^{33} + 514253817352197566228443742220a - 426678750962926081341387140032 x^{32} + (-223803051580183212520439067032a + 72311123237992807253679013536 )x^{31} + -333567245863472185640259305676a - 82537439545934439431497219980 x^{30} + 447471468867190105449380305344a - 271054645305761658185603265640 x^{29} + 345949299069271144622187587008a - 558004204956825158858699379656 x^{28} + 349285548539628939257024533640a - 227368167521759468615972479240 x^{27} + -508774250025146016992335820456a - 309805271949051122886740589748 x^{26} + 307200994276516164914317750688a - 369170270927617443980601855552 x^{25} + 599388016576568997892454294178a - 201224853676191985425867500348 x^{24} + 633174345939087096799341280104a - 556847377802765182968750044680 x^{23} + (-28657471177818040640088675036a + 103790925262221958016344216032 )x^{22} + 66262749488577959413227656000a - 486689861644314234349564700432 x^{21} + 315030593234234268934801152484a - 441633844221296944979687820160 x^{20} + (-118177743632379515464700412204a + 483413729430514924614312999912 )x^{19} + (61450897683912687038445405928a + 522370172896828097740496779920 )x^{18} + (631662288536386231056279177096a + 488925971581552075294465453400 )x^{17} + (-522704037559270752599629417120a + 112689280197489168365964175480 )x^{16} + (-526984354448762455893189958528a + 337714745330951593669656281632 )x^{15} + 622171864578714802837691772876a - 136583531263229515281225497408 x^{14} + 425863432500916135685140108692a - 385021315936284506288944715888 x^{13} + 16287526301341675692076732856a - 340415018453868973137168978452 x^{12} + (-474586425037618782582151723208a + 453251316152014791122507161952 )x^{11} + (389707692392066854442318659264a + 436985893087824961335460501600 )x^{10} + (495521908378969839158644988928a + 479465141973453746208285436568 )x^{9} + (169657120279263213084578027264a + 601150146800788437354162398064 )x^{8} + -420058215694129333866452133424a - 207390821625565101366817316816 x^{7} + (386552026021753169550875151108a + 425517640445322502676013168768 )x^{6} + (-328576810189053415795121074568a + 320793042630923937374744596704 )x^{5} + (-374441003395407999456984264192a + 18921284178868213209339072 )x^{4} + (64941839622388589170770437528a + 129702853102865990848203703264 )x^{3} + (175600702452820855112040024708a + 176818891210891306627809252632 )x^{2} + (141541292556866202621789333192a + 308006037450134655998396556272 )x - 244085758138100076858568380636a - 405185678037733123364615792638 \)
← Back to 2.1.2.3a1.1 Summary