← Back to 2.1.2.3a1.3

ex.24.10.1.131_259_387.c

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
\( v(N) = 10\)
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 \(x^{2} + 90546471444873528678335943241b^{4} x + 3b \)

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

Inertia Polynomial

The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + (-403012332264745202745056448560a + 182443817911977537905478881992 )x^{47} + 615043576039777718645168973992a - 298538008392114548018064720068 x^{46} + (-483281879565722522102154798444a + 158106611240021481520624802480 )x^{45} + -189622715445331275212721122360a - 466521132279161684309895283568 x^{44} + (-544469460086048043211433947976a + 84865090057254059784347889712 )x^{43} + (-33455351432390808464977721396a + 342602886477256819797942738220 )x^{42} + -171144196098917208502874137256a - 97272738586537143105159094680 x^{41} + (-619838250542181532374331762104a + 332567351139211238776928298188 )x^{40} + -317829978705549067622224342632a - 435616277660519309463623319776 x^{39} + (276728483624851775537669719784a + 569809213520888269859876295048 )x^{38} + 604655335022852521590212369916a - 323721842837553143855817801192 x^{37} + 152569021410706407151502827932a - 431107926635745116780357928964 x^{36} + (399230969746517911213351526976a + 178789527361581202843169671872 )x^{35} + -605852330368388005354612068484a - 279562140140637586913905822032 x^{34} + 423864918333564822730507963968a - 312087306054351379367365872520 x^{33} + (464994958333164091115196239152a + 509968313853439593602309441632 )x^{32} + (-243183219023759736923083133568a + 574409265843838496631799160960 )x^{31} + 497910630545457389461980662928a - 156616839227682300749652429376 x^{30} + -487833976298992436529737653824a - 345461585089470747712640425808 x^{29} + 465769577789974807274282037512a - 505726525362697443808932397520 x^{28} + 57017481931101789981463982816a - 183159280350859667785281939872 x^{27} + -586653960610939878833299489664a - 457685300249174324658241673400 x^{26} + -70211210782925020540319989288a - 170581246761921466754471937184 x^{25} + 160209212506197681823969059042a - 407893246173189941887100206188 x^{24} + 334346902242973174841106050264a - 341539718119589087547574544528 x^{23} + 8476260778882768279650564188a - 338989225277689705200047575176 x^{22} + -620606289955206599749071006104a - 27434737631372559095720577944 x^{21} + (358287745511984881387838996624a + 120141244268586960804307944512 )x^{20} + 573514180312209514387981724256a - 129649159773026203906475904816 x^{19} + (-260509178138969706040807002100a + 61436538965437074814457749232 )x^{18} + (401912290690948525201821159864a + 277190081206719604350378231104 )x^{17} + -476519440973060649802326531396a - 19863945948039503663414695864 x^{16} + 454475364821265917035871788224a - 36437764304457865509413886368 x^{15} + -345460869621494338220555870680a - 592543710238707860769203250528 x^{14} + (575884110251228324756547015600a + 405011003665445408524658232328 )x^{13} + (-507590113541272592969131679064a + 162499657576028405612256156848 )x^{12} + -67330680275844347107899132912a - 605529332557630525438536093504 x^{11} + (566727687681467062758855350912a + 264179321293682968720259680936 )x^{10} + 532543957968619512478477585048a - 234094463034061058763673370864 x^{9} + (-22664055215448235576449322416a + 536070123530943317056302231136 )x^{8} + 385863862084125290945137258912a - 591517748955063709890231927696 x^{7} + (612538476558710589600471429840a + 228804936598106667270613162240 )x^{6} + -388309621108044380493909650384a - 519678773058804336960494185856 x^{5} + 68971665920602256123582240048a - 45663125292347991724781712624 x^{4} + (-92648051306519802531266784144a + 346364028290998605764729773152 )x^{3} + 127014274673123762653575312696a - 601102115746529941627971440336 x^{2} + (492405630412013944032127617136a + 251166599607939776164008178704 )x + 162164899530462613352497269076a + 237369297000311557122739605470 \)
← Back to 2.1.2.3a1.3 Summary