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ex.24.10.1.33_67_101.d

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 \(x^{2} - 211275100038038233582783867563b^{3} x + (-105637550019019116791391933781a\cdot b - 211275100038038233582783867563)b \)

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^{ 2 } \\ \chi^A\left(((2a\cdot \mu_3 + 2)b^{2} + 2\mu_3b + 4)c + (2a\cdot \mu_3 + (2a - 2))b^{2} + ((2a + 1)\mu_3 + (2a + 1))b + 4\mu_3 + 3 \right) &= i^{ 0 } \\ \chi^A\left(((2a + 4)\mu_3b^{2} + (4a + 2)\mu_3)c + 4\mu_3b^{2} + 4a\cdot \mu_3 + 1 \right) &= i^{ 2 } \\ \chi^A\left((((2a - 3)\mu_3 + (3a - 1))b^{2} + (2\mu_3 + (2a - 3))b + ((-a + 2)\mu_3 + (a + 2)))c + ((3a - 2)\mu_3 + (a + 4))b^{2} + ((2a - 1)\mu_3 + (a - 3))b + (2a + 4)\mu_3 - 2a + 1 \right) &= i^{ 0 } \\ \chi^A\left(((2a + 1)b^{2} + (2a + 2)b)\cdot c + (2a + 4)b^{2} + 2a\cdot b + 1 \right) &= i^{ 0 } \\ \chi^A\left((((3a + 2)\mu_3 + (2a + 3))b^{2} + (a - 3)\mu_3b + ((2a - 2)\mu_3 - 2))c + (3\mu_3 + (a - 2))b^{2} + (-\mu_3 + (a - 1))b + (2a - 2)\mu_3 + a + 1 \right) &= i^{ 0 } \\ \chi^A\left((((2a + 2)\mu_3 + 2)b^{2} + (a\cdot \mu_3 + a)b + ((-2a + 4)\mu_3 + (3a + 4)))c + (3a\cdot \mu_3 + (a - 2))b^{2} + (3\mu_3 + 3)b + (-2a - 2)\mu_3 - 2a + 1 \right) &= i^{ 2 } \\ \chi^A\left(((4\mu_3 + 4)b^{2} + ((a - 2)\mu_3 + (2a - 2))b + ((4a - 2)\mu_3 + (4a - 2)))c + ((3a + 4)\mu_3 + (3a + 4))b^{2} + (-3\mu_3 + (2a + 1))b - 2\mu_3 - 1 \right) &= i^{ 0 } \\ \chi^A\left((((2a + 2)\mu_3 + (2a + 4))b^{2} + ((2a - 2)\mu_3 + (2a + 2))b + (4\mu_3 - 2a - 2))c + ((3a + 1)\mu_3 + 2)b^{2} + ((3a - 3)\mu_3 + (3a - 3))b + 4\mu_3 - 3a + 1 \right) &= i^{ 0 } \\ \chi^A\left(((-2\mu_3 + 4)b^{2} + ((3a + 2)\mu_3 + 4)b + (a - 2)\mu_3)c + a\cdot \mu_3b^{2} + (3a + 3)\mu_3b + (3a + 3)\mu_3 + 4a \right) &= i^{ 0 } \\ \chi^A\left(((4\mu_3 + 2)b^{2} + ((3a + 4)\mu_3 + (3a + 4))b + ((4a + 4)\mu_3 - 3a - 2))c + ((2a + 1)\mu_3 + (3a + 2))b^{2} + ((3a - 1)\mu_3 + (3a - 1))b - 2\mu_3 - a - 1 \right) &= i^{ 0 } \\ \chi^A\left((-\mu_3 + 3)c + -\mu_3 + 3 \right) &= i^{ 0 } \\ \chi^A\left((((3a + 2)\mu_3 + 2a)\cdot b^{2} + ((a + 4)\mu_3 + (3a - 2))b + (a - 2))c + ((2a - 1)\mu_3 + (3a + 2))b^{2} + ((a - 1)\mu_3 + (3a - 1))b + (-2a - 2)\mu_3 - 3a - 1 \right) &= i^{ 2 } \\ \chi^A\left(((2a + 1)\mu_3b^{2} + (2a + 2)\mu_3b)\cdot c + (2a + 4)\mu_3b^{2} + 2a\cdot \mu_3b + 1 \right) &= i^{ 1 } \end{array} \)

Inertia Polynomial

The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + (-390846681065324389186856264580a + 221573999544615246987328313752 )x^{47} + 144900746499670100839667660344a - 627145901547669359184128261340 x^{46} + 219289338137472982430445560336a - 45351041975890276332295353760 x^{45} + (163399256000823974225019587072a + 497226677008641125702320919156 )x^{44} + -303221651167181716423529541028a - 573018254172079664214456446780 x^{43} + (495596605739280199422368502292a + 125581336616751383898503405432 )x^{42} + -296120406386142623213893930580a - 589048176488876569461967498056 x^{41} + 255011257959478106580136491336a - 288003210175243763059775317872 x^{40} + -34516878087153580177519214312a - 304990012448499175808411427368 x^{39} + (538212254929444943806424178548a + 14233433806475200203617729748 )x^{38} + -323710024293253214417498201912a - 128813852067027410593980805188 x^{37} + (-611839192319213266507044260456a + 263951088261253992139956341344 )x^{36} + 124054273168724117316737272024a - 465804570629050004759758291152 x^{35} + -31974298758251504556938012604a - 564160614790854276612270301176 x^{34} + (-105217279491915970510308158764a + 119302490736025991027773520840 )x^{33} + (588723958284780123649163462816a + 344278184537783222156183575368 )x^{32} + (277798900530477447063895046344a + 15985600276047158194925337248 )x^{31} + (162757881013627680975958150204a + 497977207129139261429863571900 )x^{30} + (-170287462592577992123430212652a + 314409745505506272740258608936 )x^{29} + -618311731630976044887233903576a - 363833618931176426015616264056 x^{28} + (459393665774596557756691819912a + 615554389908979735146432334600 )x^{27} + (604498179680281596830903105296a + 81401254643769639672686913044 )x^{26} + (-497454674926975671656177308408a + 39130939682730233733140193072 )x^{25} + (-482865269026033360289897124638a + 402312297340508128465896487620 )x^{24} + -20584295660140944493343062936a - 246916536907521282539525210184 x^{23} + (-277549638342876832075329817772a + 232676338909635014399574489640 )x^{22} + (538712316607154592487673306672a + 315953126741712806735631199408 )x^{21} + (-258235413458318156095626372948a + 546345383201250901379159690224 )x^{20} + 346267024102100223101244261492a - 625988639529940577132942570648 x^{19} + 233666106892618068906036764320a - 142795055089338544861123532568 x^{18} + (25580511892685413789137053480a + 570615340212336316950470145208 )x^{17} + -612290977736856043267093546280a - 219133019125262977900165735696 x^{16} + 273099130967000311237867336752a - 201142805265081236167506334192 x^{15} + (22062913687266088858169710556a + 397276407744792032984709754400 )x^{14} + 385025958818041868756641321180a - 175703536834307717670903613160 x^{13} + (-537600697964526766745045266856a + 217656655530967986643449461708 )x^{12} + (359768098336969706037816381064a + 445101501411218763123746970432 )x^{11} + 121633103002379888812888979040a - 262668391246181875213049122696 x^{10} + 101522833781783007665089772096a - 579237032540584355840935027800 x^{9} + -26070677180190025653521577424a - 442849133360905846635333214840 x^{8} + (131443861806609401497614618704a + 352367658397887282653583057376 )x^{7} + 258569709429510947794814448556a - 470801034713400217773930553904 x^{6} + 119995141004203843797617078400a - 377404359568698587479473454040 x^{5} + -480055379052899323069536203536a - 343055094273101408963430565408 x^{4} + (-497894182922943719849946016184a + 36906159223240272146090026560 )x^{3} + (612122306962383377685932508556a + 78276527920057398718502261080 )x^{2} + -314000299901768724676594793864a - 252207104392495847605989164480 x - 437621043388089254477597497628a - 167919206262984159646560864082 \)
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