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ex.24.10.1.127_255_383.d

Base Field
\(F = \) 2.1.2.3a1.3 \( = \mathbb{Q}_{ 2 }(a) = \mathbb{Q}_{ 2 }[x] / (x^{2} + 7\cdot 2 )\) 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 \(x^{2} + 90546471444873528678335943241b^{4} x + b \)

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

Inertia Polynomial

The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + (521376880518630356123862948208a + 609093735130520711838146867336 )x^{47} + 459381215498980563915935825300a - 380304700895057330483651796804 x^{46} + 248074677146346293236409559180a - 228651061091892190741444649984 x^{45} + -175405344038879861821360262596a - 514030079959058451617377630328 x^{44} + (10980063375087929680759022208a + 384420291792280733860658577136 )x^{43} + -190908482460851088979250287596a - 117364722674769394370784987716 x^{42} + -600882423198460621621924395952a - 243771546971666111631297372936 x^{41} + -595510899934732597781961106276a - 591086594702236184341917966956 x^{40} + 321186091711663316138390768456a - 442645565461417455441752118768 x^{39} + (-63210835925738982213012097816a + 497183791176187104872772037120 )x^{38} + 418743531520377357839222222420a - 491040463719069700934470389848 x^{37} + (-237308735395164749323059635260a + 502401671149508372473511410876 )x^{36} + (-91629018647472189585908463864a + 380205113725940472935054416992 )x^{35} + 500318221813802812046463801696a - 607523377733186703750539789232 x^{34} + (-429134100478731542499360862008a + 621242880888950457359714820584 )x^{33} + 160878199897055587283894189788a - 59601228942437355902857619872 x^{32} + -64651108550347712217603101376a - 502223909825536420707413734432 x^{31} + (167740172933680872867068847616a + 379752103504609497045421706160 )x^{30} + -624042047625242525947160857880a - 120950426571968575572499302512 x^{29} + -539326182863682064841682591024a - 477214068413928648643234394112 x^{28} + -496716313362512508701708898112a - 385653907411722314874745815936 x^{27} + -442997190727035916763133711848a - 48334354752967015062780224760 x^{26} + -464395994182195801861152217280a - 620496146526699644779038148984 x^{25} + 45629862565543267235953870a - 244950702553095029732346337456 x^{24} + -410774768821585986953724065064a - 501534193762206262203688766336 x^{23} + (513879936482918480032264629892a + 144829201243041364113804766920 )x^{22} + (-544514320659362853100903541152a + 389460219057817222905100942216 )x^{21} + 127297789034842999802864309864a - 567454638096880653153397526376 x^{20} + -453075156879136895753109880224a - 564854787274678106166386513488 x^{19} + -226350209404348161944281351140a - 66647925825487969165924807400 x^{18} + (23050899221381849653821364248a + 533932828903049631735335704928 )x^{17} + (246377502541728777832390429724a + 377408947399323428074774697704 )x^{16} + (513464439321726896531249802352a + 482067889660081762208124289056 )x^{15} + 63547623265116648134984137808a - 562632739624692098137758288464 x^{14} + -447222124068185007995026963064a - 147151399542709890144793012328 x^{13} + 357776863644266210101292001304a - 613680497671482955979694088880 x^{12} + (229597302067702752890333346320a + 629205309153889466973288122128 )x^{11} + 326052521533612350094437455512a - 22746647574737865458935048736 x^{10} + -177865027396898329420915396296a - 619544183803142046438819001008 x^{9} + 23423783778001980066216847344a - 203472545696839524299092852232 x^{8} + -338336707498980710191947599360a - 273279013818436570156665840592 x^{7} + -394385432468730755500675323568a - 86210327161637930719777160992 x^{6} + (531102244116613964170653438336a + 589429359290996127257168846576 )x^{5} + -98892459799663389446792452592a - 617087850912604565089601389504 x^{4} + -280273093968983324954405342576a - 405448093010465099367080703584 x^{3} + (151147051811009607119278707288a + 504212108366548319096541658912 )x^{2} + (497461643267282162888966894680a + 428662918817426216294825296704 )x + 144532295803769761637042331848a + 10743665344103311706328924426 \)
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