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ex.24.10.1.127_255_383.a

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

Inertia Polynomial

The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + (521376880518630356123862948208a + 609093735130520711838146867336 )x^{47} + (17161428651562091447744005532a + 178977560881856767122860755244 )x^{46} + (-608382232122448393199296349284a + 290335574454540967366816115680 )x^{45} + (158655808779661758145975235956a + 568534551641227425577263959816 )x^{44} + (90144908039445385159287197408a + 542047347408908086982648496096 )x^{43} + (-522162818403347602237922857068a + 533924150959002288207628728732 )x^{42} + 52577429518538621620178194432a - 151190039439568973672168946168 x^{41} + (29720470199346451052068207492a + 74814930700323079617939469172 )x^{40} + -27904804796857221549073544280a - 268680003998188885205581104688 x^{39} + 85170728865158710324125620184a - 34572298365587028624702216320 x^{38} + -158671166686420221710110132716a - 359073642661898339768189068840 x^{37} + (-19739772004991650223124111012a + 88016693175756981983098334620 )x^{36} + 608383508492422828607831740664a - 429377919497388671561572939264 x^{35} + (342247329821967353987261053240a + 250001096002187085403107907792 )x^{34} + (-163337773842087475033242346712a + 407582311504429640352616304056 )x^{33} + -64668452152366065644572826452a - 563629812693338937016051429536 x^{32} + 240501366780047627027920856960a - 568562537454592505807179218848 x^{31} + (-297153224047940477553201530928a + 200837889957829558904288043424 )x^{30} + -82475511785695779111907528776a - 266669599703705006162660440544 x^{29} + 471092119082707862668296979584a - 481424638004814989754621076168 x^{28} + (-436045903197497799025875712080a + 391880082239711116143990418848 )x^{27} + (-56122507836376204496156002664a + 569247357767607499441077783624 )x^{26} + (-633296395681841275833192545184a + 8256223416323303646564590840 )x^{25} + -266468957939366155682740708994a - 292289218632459775503427918136 x^{24} + (-550121580595355562620381346536a + 341930089564424563643882594112 )x^{23} + (-528661065070988964176525581644a + 597301399422480307647069135480 )x^{22} + (-114101348612282674074391630160a + 545515106618237877817767765384 )x^{21} + 379897410090292005426276152264a - 39057766164019170292918332760 x^{20} + (-118431017478575700500403131408a + 597413219032416829892225435280 )x^{19} + -437789312220201505953710570212a - 405446581868203136522164992952 x^{18} + 80146015183290805262540511624a - 174687588611629581828217171680 x^{17} + 290373428740028381190614093964a - 582117908289509355169900281592 x^{16} + (-381242901760659723867574132560a + 156660812417163475473809627808 )x^{15} + (-326666054471112647547926801296a + 539441105789632202587326097744 )x^{14} + -336757408914172786163886054424a - 55872738264172468066240287816 x^{13} + (-448042105533289693657485773576a + 422895189710388970155918396512 )x^{12} + 564905307625825251021402520304a - 171953916394217910061879500880 x^{11} + -562042481573392482448645437448a - 278801353052898048105497687088 x^{10} + 492708128352430576467725412680a - 354190105411638857199029200720 x^{9} + -500955869697211192987456523872a - 632663728727124382604105866536 x^{8} + (-76805320698887538775476523648a + 241709747757212565885397946224 )x^{7} + (-91364042495746114317142231808a + 429488768737250066162305193056 )x^{6} + (-298660960802483926170133687664a + 263528488809487181822309961424 )x^{5} + -310698281450555989177805080744a - 546991909255001080893810147360 x^{4} + (523990787998643265184620096912a + 351125448113514402445648888768 )x^{3} + (202155800982227342838373617016a + 594745398940443985055078620992 )x^{2} + 212490610094456305133584512200a - 72850688497488269293809301856 x - 406629896266002516930669301184a - 414768402859234443849829051942 \)
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