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ex.24.10.1.131_259_387.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 \(x^{2} - 211275100038038233582783867563b^{4} x + 3b \)

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

Inertia Polynomial

The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + -339930483393818779143136130224a - 592799869674995720243091701464 x^{47} + (-58849846085723486081179155100a + 817592388994242655910056996 )x^{46} + (253556629018097427901270359956a + 23102693989750490293063068688 )x^{45} + -304810512465468089322839431356a - 36080168992939493092233045904 x^{44} + 490693587026058403350668277640a - 491962398273150821360981859984 x^{43} + -472135972902996704835884607708a - 523665207273823297953980804884 x^{42} + (-257985498616033536079737727160a + 101294867510299753478274086680 )x^{41} + 154089232737271204045935027400a - 42036184935815021126115946708 x^{40} + (-560939626599511766953306395928a + 280441649769832210320838374416 )x^{39} + -497309151289737832596349707112a - 173536762239762348967909293024 x^{38} + 339078586053936983429116934412a - 497147754471379781580469970880 x^{37} + 485568641640451144743270481256a - 220975119747494018137852626700 x^{36} + (137827732145652596580804141008a + 94175605272187322007898822272 )x^{35} + (-284014030014391641828594102220a + 379614863267142464805667321872 )x^{34} + (188664461797247627789421407688a + 466288653685116272230302780048 )x^{33} + -623075053583611418852153104668a - 462227882288920262254975039184 x^{32} + 244895937772611518210288339264a - 486350614688694075966422581344 x^{31} + 494955308897594021303060494440a - 37323496049979248259649703328 x^{30} + (-344018248913408442235638186208a + 476941907257849735243233761952 )x^{29} + (-223901028780563275341127057436a + 436428743019322443093345531504 )x^{28} + -393076835289626606912019700640a - 571017361759425274102977640912 x^{27} + 130126163278088749297964543624a - 293233393546229695723713334536 x^{26} + -433124505976850969919550388664a - 574192283485800441096609835616 x^{25} + -18074589034931854270131135262a - 161268267944886002783332314052 x^{24} + (-172826484310219018850651895080a + 294831302532719942090165986000 )x^{23} + (568856945379926361151454691540a + 128878192908639588268140855360 )x^{22} + 195638626409257323238992437960a - 282995084477497435192928311256 x^{21} + (-461991475267747064034140625752a + 256748281982613034814956145144 )x^{20} + 183922727276045978074044985328a - 424517969766417340687197748432 x^{19} + -469315839201821259488865518068a - 622460989798966493534473180944 x^{18} + 490441337721139673003735497672a - 583946903534281332342825415520 x^{17} + -285989983146369776063592980260a - 42411304858402645055073345592 x^{16} + -84934691630367116736565011600a - 483878516884069751264438463840 x^{15} + -51495302260065913286848718896a - 191188192609740116958684307920 x^{14} + (-423896299705752749856237793240a + 75317536485832460356418678744 )x^{13} + -433488909084263657782883405800a - 176461780739991865356101254616 x^{12} + (62080451107557423125760032112a + 292309310679457833068648180032 )x^{11} + (-462032427081528621477351721664a + 204307202365574776015989251096 )x^{10} + 481909859611876276943019845360a - 365050228693303718483831227632 x^{9} + (-360274428459970888032844508552a + 159465883056935713686543837528 )x^{8} + 628902577772482606987773570496a - 2099101055371293484159301072 x^{7} + (-405450509336622757923479394976a + 280009196434965297812956670368 )x^{6} + -609008151482948288199404684528a - 619455192178662184583557721216 x^{5} + (349491451286954173718304316408a + 65369874519603405231219982728 )x^{4} + (-140014512463857238755848786864a + 133750461463282600560904658624 )x^{3} + (496507141582937979078822196840a + 550621532403262655511841935024 )x^{2} + (26043885050388625345744514944a + 311477014718253485581320049328 )x + 284415369172529610910406538836a - 14981584550678915577548512698 \)
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