← Back to 2.1.2.3a1.3

ex.24.10.1.127_255_383.b

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

Inertia Polynomial

The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + (-541732284167565660751361729168a + 156839231132951266892244744136 )x^{47} + (-205274770387786192794731028856a + 482384302658251949594931261732 )x^{46} + -367350772035693866358641145892a - 425950318458377073310138997328 x^{45} + -201997374728495525229109664344a - 83373795290939991501135428576 x^{44} + -242612386901128224142533378464a - 105747678870997160568960361936 x^{43} + -129741688366831857785295914116a - 542960334658565262095909655028 x^{42} + (97914008972859165620408725136a + 429760668911354504632678598920 )x^{41} + (582502385447309058334082065144a + 629442407081653319611565924284 )x^{40} + (253961663861103532657497781128a + 328285169973397928850232270240 )x^{39} + -235038220418339405058255279024a - 555546915453049082159640511800 x^{38} + (171540064425209000494729583772a + 154725636177322352769805202608 )x^{37} + -62065615866605203038558300552a - 352205694719292216555058376164 x^{36} + (340268711838128027228436719880a + 11468114504677880117503024288 )x^{35} + (626472046352475768873648357800a + 415202352930003866302034380136 )x^{34} + 498173752988380717089115832424a - 546881604747252800853893465376 x^{33} + (-266575457976674341450518922304a + 17318527009051931641906542848 )x^{32} + -416439169550592723807523060096a - 272330251848310141568327051904 x^{31} + (437410722732896519582718003320a + 257407945513102905796340258720 )x^{30} + (350215120389059825824598411464a + 192064379774305419569829352688 )x^{29} + -374028732803216462991359445108a - 47254980776480973316665182944 x^{28} + -240207863971916134080145314144a - 176063168730364751305634613840 x^{27} + 40693640560049089310078303624a - 143738316622128811374044110968 x^{26} + 174566192974167169543283167624a - 474829195725371235456734013560 x^{25} + 19721610178282254961967674174a - 543164380924916246008691017576 x^{24} + (-340485778512415424229642969096a + 124541411156689478512287480992 )x^{23} + 39717354132657653559003105300a - 240046667214875683421789715824 x^{22} + (-492182193685059025480196253520a + 366087868619603815385445839912 )x^{21} + 197293643938557335456923605616a - 172992268628947614432362482240 x^{20} + (-347475322608586987228555689680a + 328368192338241457526542808208 )x^{19} + (-620685720675281995539605986436a + 155443713985943907450460141208 )x^{18} + (-182048103729789897471609124136a + 186771932796608641330785166848 )x^{17} + (-328313830715893221674535814188a + 217681918730114433899743704112 )x^{16} + (-445443661719153838553215778144a + 14027175207122401293360624672 )x^{15} + -146090868857934362982312926264a - 424905239470071963744730485024 x^{14} + (-230178999619503539789778317520a + 461267018670999880338383542824 )x^{13} + 470389917680551460045925129784a - 390453740385726291385227110504 x^{12} + -399940206316374389127959585744a - 381705468179936004714668287504 x^{11} + (-33770782570514291741278526064a + 265937975411463410946623594176 )x^{10} + (33956399287485310944085193072a + 184621133923107441052855411952 )x^{9} + (-608400713190458056098461820944a + 49790469495922827947572652608 )x^{8} + -403879103015926682677882868832a - 8554257615437478256376265744 x^{7} + (274369825343688420079361257216a + 120980089682423231839075637568 )x^{6} + 281499571145313175159022198768a - 528254996176853644439946067568 x^{5} + 286566990174061682963201890896a - 436880856563436275635105300472 x^{4} + -277086312128689422421374503760a - 53493484451042741708973335392 x^{3} + 82886287663248260729696190088a - 150268827250843356765458138608 x^{2} + -251831647085424697781376472664a - 97721120785025281514667364592 x + 339729379411275471791016734520a - 378023943035121081461528229302 \)
← Back to 2.1.2.3a1.3 Summary