← Back to 2.1.2.3a1.4

ex.24.10.1.33_67_101.a

Base Field
\(F = \) 2.1.2.3a1.4 \( = \mathbb{Q}_{ 2 }(a) = \mathbb{Q}_{ 2 }[x] / (x^{2} + 3\cdot 2 )\) 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 \(x^{2} - 211275100038038233582783867563b^{3} x + (-105637550019019116791391933781a\cdot b - 211275100038038233582783867563)b \)

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^{ 0 } \\ \chi^A\left(((2a\cdot \mu_3 + 2)b^{2} + 2\mu_3b + 4)c + (2a\cdot \mu_3 + (2a - 2))b^{2} + ((2a + 1)\mu_3 + (2a + 1))b + 4\mu_3 + 3 \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^{ 0 } \\ \chi^A\left((((2a - 3)\mu_3 + (3a - 1))b^{2} + (2\mu_3 + (2a - 3))b + ((-a + 2)\mu_3 + (a + 2)))c + ((3a - 2)\mu_3 + (a + 4))b^{2} + ((2a - 1)\mu_3 + (a - 3))b + (2a + 4)\mu_3 - 2a + 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 + 2)\mu_3 + (2a + 3))b^{2} + (a - 3)\mu_3b + ((2a - 2)\mu_3 - 2))c + (3\mu_3 + (a - 2))b^{2} + (-\mu_3 + (a - 1))b + (2a - 2)\mu_3 + a + 1 \right) &= i^{ 2 } \\ \chi^A\left((((2a + 2)\mu_3 + 2)b^{2} + (a\cdot \mu_3 + a)b + ((-2a + 4)\mu_3 + (3a + 4)))c + (3a\cdot \mu_3 + (a - 2))b^{2} + (3\mu_3 + 3)b + (-2a - 2)\mu_3 - 2a + 1 \right) &= i^{ 2 } \\ \chi^A\left(((4\mu_3 + 4)b^{2} + ((a - 2)\mu_3 + (2a - 2))b + ((4a - 2)\mu_3 + (4a - 2)))c + ((3a + 4)\mu_3 + (3a + 4))b^{2} + (-3\mu_3 + (2a + 1))b - 2\mu_3 - 1 \right) &= i^{ 2 } \\ \chi^A\left((((2a + 2)\mu_3 + (2a + 4))b^{2} + ((2a - 2)\mu_3 + (2a + 2))b + (4\mu_3 - 2a - 2))c + ((3a + 1)\mu_3 + 2)b^{2} + ((3a - 3)\mu_3 + (3a - 3))b + 4\mu_3 - 3a + 1 \right) &= i^{ 0 } \\ \chi^A\left(((-2\mu_3 + 4)b^{2} + ((3a + 2)\mu_3 + 4)b + (a - 2)\mu_3)c + a\cdot \mu_3b^{2} + (3a + 3)\mu_3b + (3a + 3)\mu_3 + 4a \right) &= i^{ 0 } \\ \chi^A\left(((4\mu_3 + 2)b^{2} + ((3a + 4)\mu_3 + (3a + 4))b + ((4a + 4)\mu_3 - 3a - 2))c + ((2a + 1)\mu_3 + (3a + 2))b^{2} + ((3a - 1)\mu_3 + (3a - 1))b - 2\mu_3 - a - 1 \right) &= i^{ 0 } \\ \chi^A\left((-\mu_3 + 3)c + -\mu_3 + 3 \right) &= i^{ 0 } \\ \chi^A\left((((3a + 2)\mu_3 + 2a)\cdot b^{2} + ((a + 4)\mu_3 + (3a - 2))b + (a - 2))c + ((2a - 1)\mu_3 + (3a + 2))b^{2} + ((a - 1)\mu_3 + (3a - 1))b + (-2a - 2)\mu_3 - 3a - 1 \right) &= i^{ 2 } \\ \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} + 51831712670854041786239427056a - 579427966205952858256958650032 x^{47} + -154208360813144825604797511028a - 249724907549995791825828069716 x^{46} + (52862230089846087097219148124a + 245219443508106459134513119440 )x^{45} + 486514513551447705875968236668a - 144191869309406554760671977564 x^{44} + (-581220734616732329108128464820a + 198438654169498634452540499060 )x^{43} + (-267435581175139999527596118364a + 617022162379562969545343762288 )x^{42} + (-382622554705454939726004683640a + 416966312761809307049969895656 )x^{41} + (-503193549146501135791765396876a + 300511292806337428810292502232 )x^{40} + 355957002149289177130809848928a - 405838319209385975545037117920 x^{39} + 66086387085685962484081957684a - 214684462010286321394317280164 x^{38} + (610406674592193276405477007864a + 413435761907463926045603796516 )x^{37} + 569339719540659564529551339580a - 347129314435687046995333067864 x^{36} + (-332778688487807058568539973768a + 458624846446964610794051351528 )x^{35} + -620483997447824600417107537524a - 140043603624861474357558925120 x^{34} + -550727065250649776069673638756a - 123267217955675971950128290472 x^{33} + -361119826212739383267192956652a - 348869078867371724078931849160 x^{32} + -134221774373511062366603279540a - 111349559335657805566437334080 x^{31} + (-221137988875937131349015772660a + 575392915020246312097275324588 )x^{30} + 31727757581857144476976168876a - 60621429034538307808587787072 x^{29} + 321535055382525396918128226960a - 73419936749481681054355598216 x^{28} + 18411607458901316658901046600a - 294552712492899140246539739064 x^{27} + (-408370242867018788055103244968a + 96156501659712560096890945508 )x^{26} + -71061055787324645846264025756a - 550669803945423303333692956200 x^{25} + (132869022188356695940483694114a + 494826104300373817931969815364 )x^{24} + (58964938056461056911242863280a + 330074619366228248975655283280 )x^{23} + 631796653413749515673166531244a - 374892569626405485253519103936 x^{22} + -136119800708155271483562482744a - 443173539836568933348939680888 x^{21} + (-535877101731211363115871384524a + 601198737599680058600100572392 )x^{20} + (461867844840748219329338543332a + 460912263139994693439297481944 )x^{19} + (-26552704723933351100546525504a + 355233795521226025234493280920 )x^{18} + 250878304512079434590629684272a - 571290654481955032496178452176 x^{17} + (221070373607799006759495160280a + 12369825123650508749235476072 )x^{16} + -68815902495715181942581707464a - 345027441785348383356925060304 x^{15} + (533343141039382776929319111844a + 8363878772399313085046945792 )x^{14} + -443925972521818348836363092260a - 210259494888007999447477422472 x^{13} + (301434511867524333911327488968a + 1948420290314735399188532900 )x^{12} + (-227364609354673374167457650848a + 349484594075712727767170169952 )x^{11} + -196591888213205416401484932696a - 619782635575094300964116254392 x^{10} + (409758864326013128065594003792a + 553558194807407592817871515480 )x^{9} + (-445378823924407878110244690648a + 117755795489237271939366338048 )x^{8} + (237048672437275816448248356960a + 524611889474435797545331886136 )x^{7} + 325993660736492499789883253804a - 300229259778004803570697037744 x^{6} + (307175168031109016007474427032a + 199420462360390319190604474472 )x^{5} + (290201059715245449603879214944a + 603059301692857003462702093088 )x^{4} + -317856614363720756443640518984a - 70160348746735420427323334480 x^{3} + (521793447727621072128382049436a + 568195775058411673587728673944 )x^{2} + (-116381407231151442317386037832a + 98596233426761554910226933752 )x + 328993341026727045575957083068a - 534106155941761045686419732882 \)
← Back to 2.1.2.3a1.4 Summary