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ex.24.10.1.131_259_387.b

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 \(x^{2} - 211275100038038233582783867563b^{4} x + 3b \)

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

Inertia Polynomial

The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + -463934126403983049750416699568a - 266935392037346070251220287384 x^{47} + (221163716997453324130759388960a + 379025724793669713102052447004 )x^{46} + -346312449995460594298136222156a - 78604451975223604391356337680 x^{45} + (134452322152839161666111408232a + 470736731847143256387105439792 )x^{44} + (-352776916746008184772325139016a + 618010199923454139269389920608 )x^{43} + (379956798201579993199493864956a + 102116506473432633810320844956 )x^{42} + (332315957429354943738850898376a + 270296952883132967399676711624 )x^{41} + -230144174459608566603450123336a - 312616521816000508076104783532 x^{40} + 398333976295773350760909500696a - 412013837049647417237706345824 x^{39} + (-315674766582498706807383869480a + 141012962012698423572481883832 )x^{38} + (-524329598370062391149433794580a + 136230485200508626030451058344 )x^{37} + (340504293749779562473158193796a + 406110248010941564930089615212 )x^{36} + (176462652999553365951509441232a + 233575542606957043385949157024 )x^{35} + (-604103812530991340722980063756a + 454566911193423861774485360224 )x^{34} + (-6101987796684238395208509552a + 424343573495068348426410605736 )x^{33} + -494342077212393921994190337712a - 373956050177854828914630128480 x^{32} + -548759813842130636076372250112a - 255793179437312857564672144640 x^{31} + (-494791406939800648050761723728a + 598677950757996473219181720656 )x^{30} + -340449328834789728599245081424a - 142048784156702909625004501024 x^{29} + (269613562438210400370017643544a + 496499873331681411471555137208 )x^{28} + -156369001849808719407305890384a - 600228018444167375166993213344 x^{27} + (391165051529793860813598091904a + 159373799442745792019080670968 )x^{26} + (68711190718564848663364903352a + 353057938977460944528764561456 )x^{25} + -229439740321353971524917349206a - 38755781290774424368293627332 x^{24} + 583557146122134484542939389112a - 603206463867117707038229966224 x^{23} + -321160337083203213392764080500a - 457492184352625370945769945016 x^{22} + 519137279413065229684114901112a - 189358173721517068484734918584 x^{21} + 374121709143012476917261701456a - 370950379758008745917464975840 x^{20} + (158656871253725675509450972112a + 626868691361106306412541767056 )x^{19} + 466180928063793870522743460652a - 539537680298555425843699544864 x^{18} + (555225439239527664441324910840a + 440730352566790541068864321216 )x^{17} + 309311527895641597413005254916a - 270163399096652083439948208216 x^{16} + 525031938339682405204251591616a - 432982803003513601631087609248 x^{15} + -229602622349098001448801231272a - 442590088824990642404010968864 x^{14} + (-141025350607292430837190537072a + 68127902721765171305342767496 )x^{13} + 380391350681029592551925186472a - 505779279354925301655158710144 x^{12} + (-387673299876146860157783292784a + 608055018405366763730479718688 )x^{11} + 294871819121128971519664807904a - 395580882990781466265625302792 x^{10} + -159519827027383366811172222328a - 379242397016246314873590023440 x^{9} + -156314931828378254766380294064a - 524880865537443531316825770464 x^{8} + 455875577199431998634338291168a - 204379125557817237687821846224 x^{7} + (-187258997185790515841715940672a + 90016108654275354763568521920 )x^{6} + (-461391271776398768620363655808a + 17617162041703546923520682240 )x^{5} + (597374438132152614868818245144a + 113655837396615417772104928192 )x^{4} + -498459168699080325755190426256a - 566804834413887818767398827008 x^{3} + 612447988760244667502202307336a - 477264140938566102220545214608 x^{2} + (-442559287395338326936566091488a + 200879301080864755675416153392 )x - 79656811269750209481882142628a + 334245556382925039275110044950 \)
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