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ex.24.10.1.131_259_387.a

Base Field
\(F = \) 2.1.2.3a1.2 \( = \mathbb{Q}_{ 2 }(a) = \mathbb{Q}_{ 2 }[x] / (x^{2} + 5\cdot 2 )\) 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 \(x^{2} + 253530120045645880299340641075b^{4} x + 3b \)

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

Inertia Polynomial

The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + (-117468031358374316604206598064a + 25747429534619484384242546584 )x^{47} + 202042141573270763127573281356a - 252747983682439133374972213216 x^{46} + (-514483830964553814885924988684a + 115106537318965511666088011312 )x^{45} + (71871278732290331609711881992a + 601445606104989590577398389916 )x^{44} + (83049452217360362741810531640a + 4838383295947048767885488768 )x^{43} + 265214817007970822801349641300a - 567146233962413562361915544692 x^{42} + (-614421928367887435774156992752a + 307974349550182085678925665248 )x^{41} + (241271004721003525112576818668a + 8345152437051787120623845172 )x^{40} + -155087421113247330327555371304a - 134792840784197260643902751792 x^{39} + (-482470711770023505491374975220a + 448454784983770218484889791528 )x^{38} + (130427323147102164023079636084a + 207564938584518116845914050696 )x^{37} + -79453536264746255574171543340a - 192880627954132460679382166548 x^{36} + -92003142438310503968734241136a - 320957188969131887886063714112 x^{35} + -65738027431881651145593171436a - 145869483305386571098491944304 x^{34} + (578021483006125655105111082920a + 318851342292455027487686596080 )x^{33} + -323238942342876814724153342432a - 223944948478049629331946413048 x^{32} + (445404481065916956881577979008a + 604743265582663098653042030848 )x^{31} + (29317667916335714291970986512a + 79468959865824339430050438288 )x^{30} + (119814912067683022154728895856a + 318100773957911879000272283528 )x^{29} + (-113409319694896586558136052768a + 491546516713989782080941866488 )x^{28} + -442436346154158977663804426016a - 189893028213171146766995857040 x^{27} + -551041889377475210830411641380a - 434316942230624092844928922160 x^{26} + 326271775805641362134725262368a - 514722946332437462175512745688 x^{25} + (-222677201766870710320606284006a + 219807108781086058081249048356 )x^{24} + -361494888415340985050459046968a - 373361346318887660573812842704 x^{23} + 101916229418486579369294026640a - 405817061839635209419702142648 x^{22} + 84235360081052608945461230904a - 80367597093380498587005060744 x^{21} + (-482660475189425738709602375700a + 346813740764342200217306581064 )x^{20} + -472061191111507680679683592992a - 108251079666308432944411796752 x^{19} + (629000014224614155095536582060a + 608486443640141576768392288144 )x^{18} + -69518706644042792326541076256a - 629656472818419088530223955728 x^{17} + (2219851856529341276266631236a + 1514873298198502384604404464 )x^{16} + (155685175039729187232303132656a + 63880040163771740885358005056 )x^{15} + -223908841107609781932790775768a - 108433631927965943801318830136 x^{14} + -479557419239482281298767997872a - 83651092544123635124014804152 x^{13} + (-328395446807440084085061814000a + 454838276753340599614381935264 )x^{12} + (-445251682880366385603318109584a + 395439031217943372911994347936 )x^{11} + -62994396606416368799389634608a - 344926675058975115549613703224 x^{10} + -50795768452338760614326894080a - 625894451261645981261799955600 x^{9} + -479146898763121292047295031320a - 302940263517751071145611873440 x^{8} + (333030013839319313645572751008a + 372883670672716281599194637776 )x^{7} + -168361015659461710451983250368a - 610283105861274908549028264704 x^{6} + -535057828114533340226028506008a - 57133607907584750797451947440 x^{5} + -225539066806599559374909208520a - 536971335801628871577908433200 x^{4} + (396327210647704383423871926320a + 425973861526725557339171523968 )x^{3} + (-25662035920540271285305692536a + 142014526454006200458211873704 )x^{2} + 282872816829366939589692864632a - 628483572705774267572616790064 x + 362689949403832045216219752364a + 131269007859283642190007809842 \)
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