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ex.24.10.1.131_259_387.c

Base Field
\(F = \) 2.1.2.2a1.1 \( = \mathbb{Q}_{ 2 }(a) = \mathbb{Q}_{ 2 }[x] / (x^{2} + 2 x + 2 )\) 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 \(x^{2} + ((a + 1)b^{4} )x + b \)

Underlying Character

Character \(\chi^A:\mathcal O_K^\times \to \mathbb C^\times\) with the following properties:

Order
4
Conductor exponent
18
Values on generators of \((\mathcal{O}_K/\mathfrak p^{ 18 })^\times/U_{\mathfrak{p}^{ 18 } }\) :
\(\begin{array}{l} \chi^A\left((-\mu_3 + 3)c + -\mu_3 + 3 \right) &= i^{ 0 } \\ \chi^A\left(((a + 4)\mu_3b^{2} + 4\mu_3b + (2a + 2)\mu_3)c + 4\mu_3b^{2} + (2a + 3)\mu_3b + \mu_3 \right) &= i^{ 0 } \\ \chi^A\left(a\cdot \mu_3c - 2\mu_3b + 1 \right) &= i^{ 1 } \\ \chi^A\left((\mu_3b^{2} + (2a + 4)\mu_3)c + 4\mu_3b + 4\mu_3 + 1 \right) &= i^{ 0 } \\ \chi^A\left(((2a + 2)\mu_3b^{2} - \mu_3b)\cdot c + (a + 2)\mu_3b^{2} + 1 \right) &= i^{ 2 } \\ \chi^A\left((((2a + 2)\mu_3 + (3a + 2))b^{2} + ((2a + 4)\mu_3 + (3a - 2))b + (2a - 3)\mu_3)c + (a + 2)\mu_3b^{2} + (2a + 4)b + 2\mu_3 + 1 \right) &= i^{ 0 } \\ \chi^A\left((b^{2} + (2a + 4))c + 4b - 3 \right) &= i^{ 0 } \\ \chi^A\left((((a - 2)\mu_3 + (a - 2))b^{2} + ((a + 2)\mu_3 + (a + 2))b + (2\mu_3 + 2))c + ((a + 1)\mu_3 + (a + 1))b^{2} + ((3a - 3)\mu_3 + (3a - 3))b + (a + 1)\mu_3 + a + 1 \right) &= i^{ 0 } \\ \chi^A\left(((-2\mu_3 + 2)b^{2} + (4\mu_3 + 2)b - 2\mu_3)c + (3a + 1)\mu_3b^{2} + ((a - 1)\mu_3 + (3a + 3))b + a - 1 \right) &= i^{ 0 } \\ \chi^A\left(((2a + 4)b + (2a + 4)\mu_3)c + ((2a - 2)\mu_3 + 4)b^{2} + ((2a - 1)\mu_3 - 1)b + 4\mu_3 + 1 \right) &= i^{ 0 } \\ \chi^A\left(((a + 2)b^{2} + ((3a + 2)\mu_3 + 4)b - 2\mu_3 - 2)c + ((3a + 1)\mu_3 + (2a + 4))b^{2} + ((3a + 3)\mu_3 + (3a + 3))b + 2a\cdot \mu_3 + 3a - 1 \right) &= i^{ 0 } \\ \chi^A\left(((2a + 2)b^{2} - b)c + (a + 2)b^{2} + 1 \right) &= i^{ 0 } \\ \chi^A\left((((a + 2)\mu_3 + 4)b^{2} + (2\mu_3 - 2)b + (4\mu_3 - 2))c + ((3a - 3)\mu_3 + 4)b^{2} + ((a - 1)\mu_3 + (a + 3))b + 2a\cdot \mu_3 + a - 1 \right) &= i^{ 0 } \\ \chi^A\left(a\cdot c - 2b + 1 \right) &= i^{ 0 } \end{array} \)

Inertia Polynomial

The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + -458823472515593261553573209976a - 210239284682070477532715081992 x^{47} + -56229027290425057405214509212a - 30995377518546714429220600060 x^{46} + (556995156941920850551430066412a + 471825731253661676327504069152 )x^{45} + (22775114642615277027847388208a + 492211427606373976512968476224 )x^{44} + 46958041703639895404012465464a - 508976850386452671074344361184 x^{43} + -220909106795621355180376639600a - 72889106254884768071900828996 x^{42} + (-250089513739016747326212841800a + 139060883775404396426146249368 )x^{41} + -542744348988078548319090681476a - 394492878361276465738363536932 x^{40} + 628574637402828688705524499688a - 363473719843174005209299928432 x^{39} + (-259007426621626378981621653672a + 136744140803740389126344428936 )x^{38} + -288932922309686815043468221140a - 625240014408367323324336180704 x^{37} + 374939392737053050541336700324a - 288652468936618776622881489100 x^{36} + (435721060956706550761735548080a + 547225940471527753390888790656 )x^{35} + -34324174708082263567638036284a - 102731521789812246245648744120 x^{34} + -240743237932082251026738182520a - 414780993638278332235851136056 x^{33} + -53147793401041411802237895304a - 512842283395926133592833158368 x^{32} + (400421829605760577573934004864a + 287081261209392236208101697472 )x^{31} + (407003326548266823145157415888a + 473692754381061192299718192144 )x^{30} + -457842305262921964052504915888a - 300707341622052547910997089728 x^{29} + 497507609697607140537388804264a - 531459319037274304401141627088 x^{28} + (-80083522923297464911710660144a + 411572826014073959879620879808 )x^{27} + -35679543311977397149682384568a - 199294835512099622334925084136 x^{26} + -260684015486147228209173418544a - 453830075213723794687561255776 x^{25} + -353320819686899270382052519914a - 149536699195677275684383885228 x^{24} + (-21823572145015646568194854376a + 342126892559270524423956542208 )x^{23} + (-377273754903133453410844470708a + 477769283110262650042661756656 )x^{22} + (-584195231740008960591929242496a + 28674668240535361245669238680 )x^{21} + (-102574261801207752729647006000a + 6675711257488915411462571472 )x^{20} + -440479984882590144338514410720a - 115487805341376015263805204592 x^{19} + -88643656301912062538974664596a - 243384359475893682381765240632 x^{18} + -116734842910778069488210587608a - 420589260455747654331839704864 x^{17} + 434349781238141019344608547412a - 412837354546587271366021174624 x^{16} + (321478480097556787638432233344a + 324455094299733041023347885792 )x^{15} + (-589995330920864148607578762744a + 602847468200995328099862449568 )x^{14} + (-162909158104543238883975838352a + 26824338435882761782442527960 )x^{13} + (-121071071338093293063000496632a + 353200693013875432306160614512 )x^{12} + -15111680031205650780544944240a - 286524692920444791614332144704 x^{11} + (-620991859533554710769958141488a + 570869385987475337335364621800 )x^{10} + (490158827114545562332987231928a + 532436083665573057805799032064 )x^{9} + (495544458327218582567689136784a + 348498501036513364590524982320 )x^{8} + 283934449746220334949983380208a - 404558855537781003567179665424 x^{7} + (480127354134484149469394408672a + 566935324643221169819251149056 )x^{6} + -338611153468914056506515076096a - 467723418607515084964278022144 x^{5} + (549206794798024729732002472016a + 257061527546525683737215066512 )x^{4} + -76353939229511969895701192624a - 420318782418428759218967628160 x^{3} + (99184834902140474076324929176a + 532591353266546948799757025248 )x^{2} + (426863141631031229689243718688a + 355626516106128655948370488000 )x - 399820625552127793160545429342a - 303507056551806523661631609590 \)
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