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ex.24.10.1.131_259_387.a

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 \(x^{2} + ((a + 1)b^{4} )x + b \)

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

Inertia Polynomial

The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + -458823472515593261553573209976a - 210239284682070477532715081992 x^{47} + -323033337671212477443681280836a - 468941004733870954986064021692 x^{46} + 543860695098941772579768800556a - 147208817872571796904816152672 x^{45} + -529660300793089118307190511696a - 618917874056979356741238575696 x^{44} + -1724943956538109251418024552a - 37061761401106362844473740016 x^{43} + (143277192375574808494430429040a + 170023504985910414044970529692 )x^{42} + -376976129936524755396885325464a - 553397583288895942017046515624 x^{41} + 236713317537015068113002704540a - 4645598150739525375826942108 x^{40} + 92005491049697027180921551368a - 601208004197489809678268267184 x^{39} + (-40508322130446980041211592600a + 580991499936642920060591716776 )x^{38} + (51801228752921612220271007036a + 533134091007342317858603233456 )x^{37} + 367269748724575140468868073484a - 514522331930173064595082261644 x^{36} + 553042031086169638203791083200a - 114042782620502363347581965984 x^{35} + (434769082059601257957110366828a + 408009923095713866332506886968 )x^{34} + (-38618992938951610200619533192a + 545311910239006682545300424568 )x^{33} + 92270544745078180605723434440a - 142685045106304690953342401408 x^{32} + (621158608539526655135820135552a + 44800784691168047661745070400 )x^{31} + -337192005522464387348082186240a - 376938177006653292155397668256 x^{30} + -289540083224141152305271090320a - 329832082601488402737459192144 x^{29} + -261139610919484350184841447224a - 549292934876652767756678877416 x^{28} + (-69135502471796825621907329824a + 128060195772108971040743127008 )x^{27} + (-605542311071979156464003883064a + 217160612447315750961123699896 )x^{26} + (-563973426764551959438394041024a + 323041775028760068994133637584 )x^{25} + -47492480759352160742476498770a - 291569084685444820035149191476 x^{24} + -409586797975605423562815659112a - 12242886664682357453236433280 x^{23} + (-42402553924494353249200743380a + 373006210916977654907142754784 )x^{22} + -551332043197142118396122573936a - 385993533268256118078114540008 x^{21} + -262906839247447977861912635232a - 481783357023930434994349947344 x^{20} + (165076581877541296639745433136a + 29558161411551108484726441040 )x^{19} + (-243705751307222032153437368244a + 317445838737635340534066804088 )x^{18} + (-39306883206235196158944418392a + 489745048442846896915828613280 )x^{17} + 553011990572096645895278739644a - 52172115313748423802158050112 x^{16} + -453523359053615552696867673280a - 402230372378808356709807751328 x^{15} + 117045570382317182841367204264a - 29804521168459373867571762880 x^{14} + (522847974966286667529095654640a + 4626588963911086940493756248 )x^{13} + 485351029896301979512862479112a - 469627347104704398606120994784 x^{12} + (220609051452889972454582297840a + 397846627963694621354068331680 )x^{11} + -632743881026903115527636808736a - 599967167603971788188586755528 x^{10} + (558040916874042608708376001096a + 231707678413471224479583980192 )x^{9} + 170220558817690077000427775888a - 523791029549439916232181225264 x^{8} + 354632238553122450823003839728a - 286854947338789430826258565008 x^{7} + 255327422822781280624998907536a - 178027228459642517460045718432 x^{6} + (-40787243427181220763563890736a + 31633212211343882482896773344 )x^{5} + 365597956406530253849385428584a - 318430553912818118212786265088 x^{4} + 458405358031886104193260858576a - 607423665067710170485798043232 x^{3} + 535702470363048521174599734296a - 123982627587899825052780203968 x^{2} + -568928197026922055753078954608a - 65331131302574582535639351296 x - 366001526931907606572480028038a + 277544520430727445631777030170 \)
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