← Back to 2.1.2.2a1.2

ex.24.10.1.127_255_383.c

Base Field
\(F = \) 2.1.2.2a1.2 \( = \mathbb{Q}_{ 2 }(a) = \mathbb{Q}_{ 2 }[x] / (x^{2} + 2 x + 3\cdot 2 )\) 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 \(x^{2} + ((70425033346012744527594622521a - 70425033346012744527594622521)b^{4} )x + 3b \)

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

Inertia Polynomial

The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + -196860485231210131548314975448a - 591439414626939433156721435672 x^{47} + (611822313781416654965178202964a + 341265736590402894044018836840 )x^{46} + -551107074436448178560533217132a - 171392811780425788820283296560 x^{45} + -257668597658341820934990464568a - 110198576886380419454427201196 x^{44} + -233694001515697249294899253824a - 350583624524961410017862274512 x^{43} + -188477115853100402710673253400a - 263081640701220849083947308068 x^{42} + (-515114690741617999295084379240a + 43083678364845513148588163120 )x^{41} + -621350095241433098293278485180a - 81269706331274404538595688396 x^{40} + (-67841944292676727048518899672a + 45362422011337501159133921088 )x^{39} + 430542977749725187477592656344a - 504706340934869607578001398856 x^{38} + -126342138072202398598506036644a - 344195877981234680804358113256 x^{37} + 591582766055081876421997349936a - 383342741400134252788449015204 x^{36} + (198326752257486369655641994312a + 74345654951637164172423308928 )x^{35} + -96710990990716444510953244124a - 91806503839019679646368156320 x^{34} + -19894991355730281296755129008a - 194606222656655279199883229528 x^{33} + 454816878122090820597690248184a - 169316984521155880029162916128 x^{32} + -394568945529258125660524578208a - 28218136530275064011684724928 x^{31} + -596511521636105703063535610696a - 180961762198434010317303662304 x^{30} + -405754221938697344021877678640a - 18071870035698917859850143496 x^{29} + (406028579417452607752265344944a + 47170257450240524592313508960 )x^{28} + (543708623074737907347468901504a + 253502215408505513574001991232 )x^{27} + 509884629938261702879884168900a - 624517743715613013513442113512 x^{26} + 471771855084292873441338429056a - 619422302725922854497663002064 x^{25} + (-310031741601636388714542213650a + 215328459201003354287132712816 )x^{24} + 92072012841856298483543633896a - 487972749209278074273267893680 x^{23} + (-51919611769353809065861492632a + 296371933416464277973376130648 )x^{22} + -209796960387365765634605123064a - 614766723751707853056805693576 x^{21} + (-97956448060798269573372535764a + 5375378300259873551421930096 )x^{20} + 135792836896830117364823112496a - 113464480317014742150626069712 x^{19} + 463390367032521438989529603132a - 520068816043219251453861644224 x^{18} + (214008874609601288253364516544a + 546930910450896894355113805472 )x^{17} + (489224882118986459234250261132a + 499823230380234019748412576280 )x^{16} + (-241293120135911421525371426224a + 69319278856198546403870741504 )x^{15} + -293956043050083058641847782640a - 268460067501645149352181247392 x^{14} + -560053298991500820605156744640a - 279955980176131092611359395192 x^{13} + (-394614280607585621663381081856a + 187382763904980313571541036760 )x^{12} + -199630719095363633919316927808a - 473617898513355429961105263472 x^{11} + -344099861004186518932355503776a - 10616761468895795293704017448 x^{10} + -301575076180841720156733011128a - 257354315555735494563930412736 x^{9} + (77120305363331571473081175296a + 99150702225213321856114049296 )x^{8} + (-423980948450709628132003504368a + 224497577435401727862128374160 )x^{7} + 346242292990851165853410281632a - 585024735526779175930556323904 x^{6} + 197394416617473643594993192216a - 303601951912485406066992295904 x^{5} + 90225771808322148519702446160a - 497167197641936736602865020864 x^{4} + 1170491824802679932811054608a - 136315768637934786163563715680 x^{3} + -33707025752571709405381203488a - 238962825464113283384445997416 x^{2} + -112394383252986991235440966720a - 106773968854571769510352732928 x + 228212645539063298172754218334a + 92537473414884976724563355122 \)
← Back to 2.1.2.2a1.2 Summary