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ex.24.10.1.33_67_101.c

Base Field
\(F = \) 2.1.2.2a1.2 \( = \mathbb{Q}_{ 2 }(a) = \mathbb{Q}_{ 2 }[x] / (x^{2} + 2 x + 3\cdot 2 )\) 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 \(x^{2} + ((70425033346012744527594622521a - 70425033346012744527594622521)b^{3} )x + ((-105637550019019116791391933781a - 211275100038038233582783867562)b + (-140850066692025489055189245042a + 140850066692025489055189245041))b \)

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 + 2)\mu_3 - 2)b^{2} + (4\mu_3 - 2)b - 2\mu_3 + 4a + 4)c + ((3a + 3)\mu_3 + 2a)\cdot b^{2} + ((a - 3)\mu_3 + (a - 3))b + (2a + 4)\mu_3 - a + 1 \right) &= i^{ 0 } \\ \chi^A\left((((a + 4)\mu_3 + (2a + 2))b^{2} + ((2a + 4)\mu_3 + 4)b - 2a\cdot \mu_3)c + (2\mu_3 + 2a)\cdot b^{2} + (2\mu_3 + 2)b + 3a\cdot \mu_3 - a - 3 \right) &= i^{ 2 } \\ \chi^A\left(((3a\cdot \mu_3 + (a + 3))b^{2} + ((2a - 2)\mu_3 + (3a - 1))b + (a\cdot \mu_3 + 2a))\cdot c + (\mu_3 + 3)b^{2} + ((3a + 1)\mu_3 + (2a - 1))b + (-a + 1)\mu_3 - a - 1 \right) &= i^{ 0 } \\ \chi^A\left(((2a + 1)b^{2} + 2b)\cdot c + 2a\cdot b^{2} + 2a\cdot b + 1 \right) &= i^{ 0 } \\ \chi^A\left((((2a + 4)\mu_3 - 3)b^{2} + ((2a + 4)\mu_3 + (a + 4))b + ((4a + 4)\mu_3 - 1))c + ((2a + 4)\mu_3 + (2a - 3))b^{2} + (4\mu_3 + 3a)\cdot b + -2a - 1 \right) &= i^{ 0 } \\ \chi^A\left(((2a + 2)\mu_3b^{2} + ((3a + 4)\mu_3 + (2a + 4))b + ((3a + 4)\mu_3 + 4a))\cdot c + ((a + 2)\mu_3 + 4)b^{2} + ((2a + 3)\mu_3 + 4)b + (-2a - 3)\mu_3 \right) &= i^{ 0 } \\ \chi^A\left((3a\cdot \mu_3b^{2} + (a + 4)\mu_3b + 2a\cdot \mu_3)c + (a - 2)\mu_3b^{2} + (2a + 1)\mu_3b + (-2a - 1)\mu_3 \right) &= i^{ 0 } \\ \chi^A\left(((-2\mu_3 + a)b^{2} + (4\mu_3 + (3a + 4))b + ((-a + 2)\mu_3 - 2a + 4))c + ((a + 2)\mu_3 - 2)b^{2} + ((2a - 1)\mu_3 + 4)b + \mu_3 + 4 \right) &= i^{ 2 } \\ \chi^A\left(((4\mu_3 + 4)b^{2} + (2a + 4)b + (4\mu_3 + (4a + 4)))c + ((a + 2)\mu_3 + (a + 4))b^{2} + (\mu_3 + (2a - 3))b + (-2a - 2)\mu_3 - 2a - 1 \right) &= i^{ 2 } \\ \chi^A\left((((2a - 2)\mu_3 + (2a + 2))b^{2} + (3a\cdot \mu_3 + (a + 4))b + (3a\cdot \mu_3 - a + 4))c + (-\mu_3 + 3)b^{2} + ((a - 3)\mu_3 + (a + 1))b + (-a - 3)\mu_3 + 3a - 3 \right) &= i^{ 0 } \\ \chi^A\left(((a + 2)b^{2} + (2a - 1)b + (2a - 2))c + (2a - 2)b^{2} + (3a + 2)b + 1 \right) &= i^{ 0 } \\ \chi^A\left((((a + 2)\mu_3 - 2)b^{2} + ((2a + 4)\mu_3 + (3a + 4))b + (2\mu_3 + (3a - 2)))c + ((2a - 1)\mu_3 + (a - 2))b^{2} + ((3a - 1)\mu_3 + (a + 3))b + (-2a + 2)\mu_3 - a + 3 \right) &= i^{ 0 } \\ \chi^A\left(((2a + 1)\mu_3b^{2} + 2\mu_3b)\cdot c + 2a\cdot \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} + (392689259821872908561012493804a + 543726715829877650579450068896 )x^{47} + (-68900367283443718404734354488a + 230048196639645632872253347420 )x^{46} + 633122889283942447655390006024a - 515953426393025279108684189752 x^{45} + (-291295616993831139396229470564a + 11758609970905489844319820092 )x^{44} + (466502103101997446472097547748a + 541760975767641988242385038908 )x^{43} + (493269229164082783678250069064a + 306559278772415990538214468304 )x^{42} + -411782594629986976167227362340a - 565969918379577410028725330416 x^{41} + (-319537605467942863823886019352a + 477471861336431783433670037296 )x^{40} + -592550499137858519186056893040a - 260175158628594231967635239880 x^{39} + (-157283270382356931515256570780a + 65196645493805574957939581772 )x^{38} + (-491631071103517832128950438268a + 212140039553059077421797787164 )x^{37} + (-460478656811848022383986023008a + 375378493881297688981873985672 )x^{36} + 133294465807800990036166168824a - 468685118472609207760480291936 x^{35} + 416709980719267470290687658476a - 553454181342477827657953285184 x^{34} + (-548475182644595545931499428980a + 469440194098795095763263411064 )x^{33} + 243621034994666776775849210832a - 136521134753236752545365513944 x^{32} + (-301556114465970543031864863400a + 528046405209935449924372506032 )x^{31} + 254704823186631817198787922032a - 430933793369466723903089051260 x^{30} + (-119388164940609111063806307692a + 58085180442780459854653275960 )x^{29} + (18791921053719015066606315136a + 172134249647706629065429782808 )x^{28} + (627809524677752240427502050824a + 539481674511503471995936555416 )x^{27} + (199683814540145209876350029224a + 205335608100415000838890665180 )x^{26} + (488945033516779296263890235480a + 130032945046111920362669606680 )x^{25} + (-391267535529009791598962632546a + 6103581448273153727096252332 )x^{24} + 623671504111285000810296048840a - 482334461436487490452687666776 x^{23} + 191380113731952431772316828564a - 131490104360051877014827369224 x^{22} + 422373225565233067262325684680a - 510752364648839053383557529936 x^{21} + (307986622784794761032216578172a + 339646714721372661890033504648 )x^{20} + -278485164722906377947605869236a - 111312602759260819655438980424 x^{19} + 99784965254253024211527485856a - 463859120229899753887667214160 x^{18} + (-181925059722826429944758324360a + 75205430747364841389704803064 )x^{17} + -326440524069378409963835634168a - 73847839218896776794433781984 x^{16} + (-86295338055967446398288379504a + 428416115662444551944569990512 )x^{15} + -455060686118238740279355009388a - 65298948820364439668452195840 x^{14} + 3981055476666299385345527380a - 84302938894413348056132503552 x^{13} + 563081184493525080704541680176a - 99804245948759374622450433692 x^{12} + (-628530765964640327967016644808a + 303191423303131915714151266688 )x^{11} + (490870229424693991203598810384a + 229832756755548200898821512616 )x^{10} + (-430899490682533241393300012856a + 46846013046997079141380677320 )x^{9} + 421878518769192113970755294024a - 609105893729375110868408931048 x^{8} + 538478756792934621759277141040a - 297876217768164929326232741216 x^{7} + -60629746990149182075743579436a - 552191290428045046404213346840 x^{6} + 527057891122411738363704684888a - 633749939400924651170696137032 x^{5} + 48274244608474667793834994176a - 107130589328920902168996797568 x^{4} + -156309234330578656665049165064a - 126597118057998733252909721792 x^{3} + (211733881448942948935798320668a + 217987825733912127505784767624 )x^{2} + 173267364706354122240345669568a - 210629484045723968307179018496 x + 310747483747736111425938137730a + 18149944541825736481893176222 \)
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