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ex.24.10.1.31_63_95.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 \(x^{2} - 211275100038038233582783867563b^{3} x + (-211275100038038233582783867563\mu_3 - 211275100038038233582783867563)b \)

Underlying Character

Character \(\chi^A:\mathcal O_K^\times \to \mathbb C^\times\) with the following properties:

Order
4
Conductor exponent
20
Values on generators of \((\mathcal{O}_K/\mathfrak p^{ 20 })^\times/U_{\mathfrak{p}^{ 20 } }\) :
\(\begin{array}{l} \chi^A\left(\mu_3c + 1 \right) &= i^{ 1 } \\ \chi^A\left(((4\mu_3 + 4)b^{2} - 2a\cdot \mu_3)c + 4\mu_3 + 1 \right) &= i^{ 2 } \\ \chi^A\left((2a\cdot \mu_3 + (2a + 2))b\cdot c + 4\mu_3b^{2} + (4\mu_3 + (2a + 4))b - 3a\cdot \mu_3 + a + 1 \right) &= i^{ 0 } \\ \chi^A\left(((3\mu_3 + 3)b + 2)c + (3a\cdot \mu_3 + 3a)\cdot b + 1 \right) &= i^{ 0 } \\ \chi^A\left(((-\mu_3 - 1)b^{2} - 2b + 4\mu_3)c + (2a\cdot \mu_3 + 2a)\cdot b^{2} + 2a\cdot b + 1 \right) &= i^{ 1 } \\ \chi^A\left((-3b + 2\mu_3)c + a\cdot b + 1 \right) &= i^{ 2 } \\ \chi^A\left(((2a\cdot \mu_3 + (a + 2))b + ((a - 2)\mu_3 + 4))c + (3a - 1)b + (a + 3)\mu_3 \right) &= i^{ 0 } \\ \chi^A\left((((3a - 2)\mu_3 + 4)b^{2} + (2a - 2)b + (a\cdot \mu_3 + 3a))\cdot c + (-\mu_3 + (2a + 4))b^{2} + (3\mu_3 + (2a - 1))b + (-a - 3)\mu_3 - 2a + 3 \right) &= i^{ 2 } \\ \chi^A\left((2b^{2} - 2\mu_3b - 2a\cdot \mu_3 - 2a)\cdot c + 3b^{2} + (2a - 3)\mu_3b + (2a + 1)\mu_3 + 2a + 1 \right) &= i^{ 0 } \\ \chi^A\left((((2a + 4)\mu_3 + 4)b^{2} + (2a + 4)b + 4a\cdot \mu_3)c + (2a + 2)\mu_3b^{2} + (-3\mu_3 + 4)b + (2a - 2)\mu_3 + 2a - 1 \right) &= i^{ 2 } \\ \chi^A\left((((a + 4)\mu_3 + (3a + 2))b^{2} + (-2\mu_3 + (3a + 2))b - 2\mu_3 + 4a)\cdot c + (2\mu_3 - 2)b^{2} - 3\mu_3b - 2\mu_3 + 2a + 3 \right) &= i^{ 2 } \\ \chi^A\left((\mu_3b^{2} + (2\mu_3 + 2)b + 4)c + 2a\cdot \mu_3b^{2} + (2a\cdot \mu_3 + 2a)\cdot b + 1 \right) &= i^{ 0 } \\ \chi^A\left(((2a\cdot \mu_3 + 2a)\cdot b^{2} + 2\mu_3)c + (4\mu_3 + 4)b^{2} + 4a\cdot \mu_3 + 1 \right) &= i^{ 0 } \\ \chi^A\left((-\mu_3 + 3)c + -\mu_3 + 3 \right) &= i^{ 0 } \end{array} \)

Inertia Polynomial

The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + (-258463008709453818939828714228a + 17930292236923456782004876992 )x^{47} + -459751522271555407905046786156a - 125344488881536071621425474960 x^{46} + 131219602833102573631179321776a - 99874131794411204418354918672 x^{45} + (407478709477236471640137426328a + 118964477676176942992747303056 )x^{44} + 348759456372301997863878541608a - 78564886378776633603861689100 x^{43} + (-15318796994106437910847810052a + 454730244391703464680945933128 )x^{42} + (378409650241515389922632403184a + 1807449738783986589972040680 )x^{41} + (-271190139550247957617624808960a + 26844307917439531822634012472 )x^{40} + 95699832396661215080595472752a - 155104064639974419073931569040 x^{39} + (-144993500115357529441473968688a + 64027334301745912975560984004 )x^{38} + -191911964704566388083833352560a - 32933881848623616605815061064 x^{37} + (-62942757189275628854169054656a + 1480892940289866001407682232 )x^{36} + -404811804490695509675027663680a - 99480134136609295135094037240 x^{35} + (59003813154560774596441572308a + 222681844204270447618243206816 )x^{34} + (403792674210903870561119922296a + 422504622569692287239651656472 )x^{33} + -218214264624375364565714968868a - 144352822257767099995077877144 x^{32} + -449361422454707338746253530224a - 531352056874875570132532184096 x^{31} + -354098227664362550415567864192a - 513835977635568832732462400556 x^{30} + 135846292451781674023437983536a - 290829744828961575510129600960 x^{29} + -361738854512265663272348011792a - 52904128119786089750468443776 x^{28} + -601122463919661427843282264880a - 146031240149489549758833202288 x^{27} + (-70345662153101930561867058240a + 184237180276412376577148625880 )x^{26} + -112412316417686717368002811008a - 60665732087745431562828241208 x^{25} + -418003101247429135695842276866a - 370744832067127239681571723544 x^{24} + -616094965382839354303323381840a - 347258859773181297605501820328 x^{23} + (-309350549278420687376081584400a + 558158955056256824425307866312 )x^{22} + (-409957704581333484575886466128a + 480571019915543860949782007360 )x^{21} + (179174030739476872021247605752a + 249955981782923264781488219000 )x^{20} + -177937108175158479349293802788a - 124421187285254847258451281872 x^{19} + (403457412427711460836953689192a + 563845209070266007793253698056 )x^{18} + (611484136062617905802811611656a + 323585053972149587729645065472 )x^{17} + (88353405979476813870631747776a + 54534819821379498546390979376 )x^{16} + (-4731504640341447273404381184a + 327661014814340312604676087024 )x^{15} + -425185140672693962509906608428a - 244214510349329768575485870208 x^{14} + (-633162161312663384020601676904a + 346663713869301669034713096576 )x^{13} + (214271346735769962557707730552a + 484851476743561539893585145988 )x^{12} + -6668199125542267866211278440a - 64709135951216443005338376736 x^{11} + -13077280996045926452074140416a - 46287050085484664391061373176 x^{10} + (420166471230873665896321976936a + 466585698477949066324673997936 )x^{9} + (16046785280875016560560107736a + 186215158141358790830250632760 )x^{8} + 211291527595928286723965583712a - 490775320496483964488981067648 x^{7} + (110426903953531310626661151428a + 510806770054326634007138284496 )x^{6} + -483486241631262906798119896992a - 126752718618052482054616108576 x^{5} + -441719579958654462747804327616a - 573338577341333668182005427952 x^{4} + (-424460125611238802109128563776a + 195795439149996932256001880896 )x^{3} + (-506270151220712294567382487848a + 229027275053514110394497736704 )x^{2} + -482164566979140735108038816544a - 332680060434743573863230912160 x - 371311985678486533618094726328a - 43498221516745298409994583270 \)
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