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ex.24.10.1.127_255_383.d

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 \(x^{2} + ((a + 1)b^{4} )x + 3b \)

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

Inertia Polynomial

The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + (-292175201196363914731666799768a + 384591559050319454541746362424 )x^{47} + 492928771270026969244208677128a - 527035499714803959951083378612 x^{46} + -557867162482788390501212535532a - 609786827956984018574709158352 x^{45} + (415789690879477818018956105996a + 186801558758378120797113675832 )x^{44} + -365179833720433294093487705632a - 532321711651533180228561539824 x^{43} + (144402407724541857687374592808a + 133885263180940408595928132348 )x^{42} + (361570621074215106994455384784a + 424934308696341060490083623528 )x^{41} + 434059359342428585436171397616a - 332719247962480342023521588908 x^{40} + -400938950356469481993653806536a - 432341555235472229160993238560 x^{39} + (609812843643669656845967472968a + 455004713498799182678311061840 )x^{38} + (-190348485996522119329271326500a + 373804197141864185406026282224 )x^{37} + -306474842448958911954128973004a - 559191967643987921863184070540 x^{36} + 586013444649936448864490187768a - 162346645576370765815807093968 x^{35} + 254178295404257898892178934568a - 138073841731263992297124145680 x^{34} + -281091811112225578867656157280a - 288988003253058342055843092744 x^{33} + (576434767797439610632908558732a + 347124271619042621984134480864 )x^{32} + (-264589119198093414847123004064a + 435835125271914932270853047584 )x^{31} + (308412097328500617638416897104a + 600655658777860275578571233456 )x^{30} + 121401101850617110196096619144a - 44060893313690976726215028864 x^{29} + (295375189240417357122485624384a + 159928447284022910247269080656 )x^{28} + -357640696056118895498987140480a - 445445116244750461248030428800 x^{27} + (310604637681867550161969374184a + 517152478431837566071678124088 )x^{26} + 392748722826648945111212706120a - 139665247579261063156616350680 x^{25} + (-116036092261570009285722540498a + 134318248122014566855956547144 )x^{24} + (-535503403450680656767670073480a + 30123405952272620639819315600 )x^{23} + 39619706728558435421489442436a - 95755942101771030977603503296 x^{22} + (67408603136202956630996481160a + 499883557727433370200226763016 )x^{21} + (-276616072902977421811482460560a + 418830730228538392548536119976 )x^{20} + -249844525348076135706287724688a - 552978762789912305467906352912 x^{19} + (-107665079252551811585535940212a + 347450477607957208518080324848 )x^{18} + (-583520668538012175977130344920a + 124541000645297776000039119520 )x^{17} + (-594646192066362228722630246244a + 530885510830455309335050917728 )x^{16} + -167957389650613494265559447632a - 46088791369202500108546316960 x^{15} + (-427100342794277810738424145616a + 517119774827856750122316569168 )x^{14} + (253501786919519593034225308360a + 500374324988219815700216728600 )x^{13} + -523555308106977037602622992104a - 628862123501587496169201511456 x^{12} + 443737193265803287335316400272a - 614769448196432284167966108560 x^{11} + (179552187639418687498109202824a + 516907924932507186013339359920 )x^{10} + (626191560597250870198302986632a + 31382487997128096419855923776 )x^{9} + (229445347528566404148468276680a + 580809498936963984452568576232 )x^{8} + -477031304346593317763173269872a - 34394645790971609332840616016 x^{7} + -63444410865463095943770866544a - 453318635363171560341253730112 x^{6} + 617898830922566337635563847680a - 502533355121801166467870278864 x^{5} + (-444569618369410430442102056208a + 608707035030311945877844727616 )x^{4} + (-163593082617513117250286831120a + 505903378523797366272146428064 )x^{3} + (597892984239208155330048139416a + 353576133156602872368905429792 )x^{2} + 89582994844722176313209204296a - 325009621337952932300387854448 x - 509336678453948895443857867690a - 134770250827911217827527420762 \)
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