← Back to 2.1.2.2a1.1

ex.24.10.1.31_63_95.c

Base Field
\(F = \) 2.1.2.2a1.1 \( = \mathbb{Q}_{ 2 }(a) = \mathbb{Q}_{ 2 }[x] / (x^{2} + 2 x + 2 )\) 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 \(x^{2} + ((a + 1)b^{3} )x + (-\mu_3 - 1)b \)

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

Inertia Polynomial

The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + (-503805321684194532527988369556a + 443308299668500445543355558952 )x^{47} + -365688808287330683732623673708a - 5433494356299150977997292520 x^{46} + (-68406992152408988719399262648a + 19448742980159579004803251544 )x^{45} + (421456340399083405234059409404a + 54830005159220835546457211960 )x^{44} + (151867216609070478985862524776a + 110227397745747670485708980180 )x^{43} + 280169159764354858873867762060a - 519087661556629210199513330536 x^{42} + (-582599228775405688188194905048a + 227994388056495897058849234800 )x^{41} + 497562461841006773402894670136a - 2426191723494512897059602312 x^{40} + 319752526313023379060500356536a - 275766281179721540188638177360 x^{39} + 590532324080447576535533965704a - 566206080766349098458681861412 x^{38} + (-28494087877008878360135051088a + 548889725232151695447584677096 )x^{37} + (36463296910314000471193164328a + 10077731333498453541014358200 )x^{36} + (-482497250086270494153001900352a + 603886478547014823488529454896 )x^{35} + -150620556208481362264540602872a - 162185581457298977421688735648 x^{34} + -115606785283018708609185284776a - 630596280137521599991482962632 x^{33} + 432517303430508687289706351120a - 19468615852358170266344905440 x^{32} + -430195102280733115261189939344a - 256687188834171147139826966720 x^{31} + 342918592561261136265459997228a - 358414021099303239526606794420 x^{30} + (469337942063972478533722059324a + 213354895214123150465099323360 )x^{29} + 229349165883262750842202584592a - 382316039716528087481684267640 x^{28} + 207312874587399718033589577264a - 128954758344050411723906438032 x^{27} + 109082654121895140613596300152a - 305326698919902439104539837000 x^{26} + (-315341464763226609738319523296a + 555038040141936030473045026880 )x^{25} + (53758250415105710061142473270a + 375536068574031465239436301016 )x^{24} + (-606652889250475576635957223600a + 293181135653681832079991374408 )x^{23} + -161809604924005743659557620560a - 604818517966510269087882228520 x^{22} + -612395949504041252177239711320a - 102430906374493120691780780624 x^{21} + -96851790790258292034778122576a - 250233480093554314563996296832 x^{20} + -8890278036598414133874313396a - 589191407506805950547486788912 x^{19} + (-361370959307842100738292014672a + 585869211637937888527799475416 )x^{18} + 131633533500621558128762226016a - 337895529817420431278648678592 x^{17} + (509934166803328978760649341736a + 516001863097340265569949840160 )x^{16} + 6320312889710919288843267776a - 250255585222056951908775519360 x^{15} + -394344520770949365373067100540a - 325370399795523916119095435296 x^{14} + (279268719148391606214105077288a + 165789527343190582063263212448 )x^{13} + (-586142420065482591068952398088a + 214344128243311154242541786796 )x^{12} + 248593100285788226926276837368a - 497815534192354012964978086384 x^{11} + (-150737613094929977119744113088a + 279925616345313927701804423312 )x^{10} + (598035332250677216091965696440a + 306355907251147320658992016656 )x^{9} + (82496303499733756328813524256a + 382729708188246456996425692400 )x^{8} + 300571053469119025763103504256a - 375412855746378522178576182480 x^{7} + (181593150626535596431165133620a + 522822847162879118290413476760 )x^{6} + 493412430549089936797631155976a - 325065486849589422695441510856 x^{5} + -126856358499584810299224483992a - 28685720009310919872916194672 x^{4} + 465479001893769469510943803072a - 410989702277537534627863423584 x^{3} + -137909942804094349072202739272a - 40952974265661539220989000896 x^{2} + -589217299657780225841626151144a - 115300514181416323975792907856 x - 18302079438189152704021277114a + 144222089379144842880408197166 \)
← Back to 2.1.2.2a1.1 Summary