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ex.24.10.1.31_63_95.a

Base Field
\(F = \) 2.1.2.3a1.1 \( = \mathbb{Q}_{ 2 }(a) = \mathbb{Q}_{ 2 }[x] / (x^{2} + 2 )\) 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 \(x^{2} - b^{3} x + (-\mu_3 - 1)b \)

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 \((\mathcal{O}_K/\mathfrak p^{ 20 })^\times/U_{\mathfrak{p}^{ 20 } }\) :
\(\begin{array}{l} \chi^A\left(\mu_3c + 1 \right) &= i^{ 1 } \\ \chi^A\left(((4\mu_3 + 4)b^{2} - 2a\cdot \mu_3)c + 4\mu_3 + 1 \right) &= i^{ 2 } \\ \chi^A\left(((2a + 4)\mu_3 + (2a + 2))b\cdot c + 4\mu_3b^{2} + (4\mu_3 + (2a + 4))b + (a + 2)\mu_3 + 3a - 3 \right) &= i^{ 0 } \\ \chi^A\left((-b - 2\mu_3)c + a\cdot b + 1 \right) &= i^{ 0 } \\ \chi^A\left((\mu_3b^{2} + (2\mu_3 + 2)b + 4)c + 2a\cdot \mu_3b^{2} + (2a\cdot \mu_3 + 2a)\cdot b + 1 \right) &= i^{ 0 } \\ \chi^A\left(((-2\mu_3 + (2a - 2))b^{2} + (-3\mu_3 + (2a + 3))b + ((a + 4)\mu_3 - 2a - 1))c + ((a - 2)\mu_3 + (a + 2))b^{2} + ((2a + 3)\mu_3 + (3a + 1))b + (-2a - 2)\mu_3 - 3a - 3 \right) &= i^{ 2 } \\ \chi^A\left(((2a + 4)\mu_3b^{2} + 2a\cdot b + (4a\cdot \mu_3 + 4a))\cdot c + ((2a + 2)\mu_3 + 4)b^{2} + (3\mu_3 + 4)b + (-2a + 2)\mu_3 + 2a + 3 \right) &= i^{ 2 } \\ \chi^A\left((((3a + 4)\mu_3 + (3a + 2))b^{2} + (3a\cdot \mu_3 + (a - 2))b + ((2a - 2)\mu_3 + (3a + 2)))c + 4b^{2} + (2a + 1)\mu_3b + (-2a - 2)\mu_3 + 2a - 3 \right) &= i^{ 0 } \\ \chi^A\left((-2b^{2} - 2\mu_3b - 2a\cdot \mu_3 - 2a)\cdot c + 3b^{2} + (2a - 1)\mu_3b + (-2a - 3)\mu_3 - 2a - 3 \right) &= i^{ 0 } \\ \chi^A\left((((2a + 4)\mu_3 + 2a)\cdot b^{2} + (3a + 2)b + (a + 2)\mu_3)c + (3a + 1)b + (3a + 3)\mu_3 \right) &= i^{ 0 } \\ \chi^A\left((((3a + 4)\mu_3 + (a + 2))b^{2} + (2\mu_3 + (3a - 2))b + 2\mu_3)c + (2\mu_3 - 2)b^{2} + 3\mu_3b + 2\mu_3 + 2a - 1 \right) &= i^{ 0 } \\ \chi^A\left(((\mu_3 + 1)b - 2)c + (3a\cdot \mu_3 + 3a)\cdot b + 1 \right) &= i^{ 0 } \\ \chi^A\left(((2a\cdot \mu_3 + 2a)\cdot b^{2} - 2\mu_3)c + (4\mu_3 + 4)b^{2} + 4a\cdot \mu_3 + 1 \right) &= i^{ 2 } \\ \chi^A\left(((-\mu_3 - 1)b^{2} - 2b + 4\mu_3)c + (2a\cdot \mu_3 + 2a)\cdot b^{2} + 2a\cdot b + 1 \right) &= i^{ 1 } \end{array} \)

Inertia Polynomial

The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + -526685287819304360435490361860a - 54771264462857055821212159096 x^{47} + -177071631828043406486575561660a - 432334881570104119903128015336 x^{46} + (228557574628637878361528192792a + 288099998459492191580195680992 )x^{45} + (-283561773668538626769498868376a + 343970715670043615691106602744 )x^{44} + -310048300965919682760989564112a - 348268999916231815085894143092 x^{43} + -403065470313156700703123246680a - 631489857053695364853459703352 x^{42} + (-10103485087148954651853581348a + 240786042286151061540394210216 )x^{41} + -306804219815149342012330338032a - 627259303311121044955870077096 x^{40} + -399687113431329522126716089848a - 244525595303105811105904213024 x^{39} + (-608072315070686425990944796776a + 459001195259530005389960664340 )x^{38} + -322851209008418302611954071260a - 313615361944137845880599537392 x^{37} + 170209431156132701044769242124a - 383731052329022191154348364160 x^{36} + -113011669908526405466763528680a - 361540903933286208260252356080 x^{35} + (323372595461028791458385705920a + 609129608946957541348306755136 )x^{34} + (-481684878216486062262232396152a + 245018796347656876716994040280 )x^{33} + 325065822910181694287308732448a - 353407523260880886508574617824 x^{32} + (105053119838254741620533428764a + 326475564682722671523698989104 )x^{31} + (237535035399012081317540110184a + 628057926520620821761518736268 )x^{30} + 386596448688662377625785321396a - 318724963852146230860331918656 x^{29} + (-123888239276257040856512023208a + 333847520003051740429500103256 )x^{28} + 442865033609247244136041155040a - 147348170827624600344487332096 x^{27} + 249899365337448958658962038632a - 617298658337342805219549344640 x^{26} + 349965247337521826791197396960a - 62792701990236114455832793072 x^{25} + (348127020388560410350686780702a + 222006194638944964227484737752 )x^{24} + -306450611418027195035866360656a - 560675580767404709744876155720 x^{23} + -480509888402893657150040129240a - 412470764520714940492535906728 x^{22} + (357505512287732833727565564944a + 459357221720090411738944816112 )x^{21} + (538997418639168055606228380496a + 53359375983100794217553046840 )x^{20} + -289208600803939986002563603428a - 386200553960550297484527038224 x^{19} + (-603858323385599970668007370800a + 261952410038157894334776983680 )x^{18} + (-548412754105262274907297889400a + 379537654714839036818502164360 )x^{17} + (301617051147643494798289468936a + 283826057626703931526555605232 )x^{16} + -532958766324362226009442926320a - 50350129736329462535226606784 x^{15} + -162799845000696753090250125652a - 257586332855999252655687591760 x^{14} + (-378598902582191277502815013832a + 540581187770993982433121567544 )x^{13} + (529218165516788247098953121432a + 551345191020366149607424745020 )x^{12} + (235223699099438086033799537576a + 600480563681254325474896935376 )x^{11} + (-148185254066360240581072456864a + 134798555920060476935589372736 )x^{10} + 239366208272153163986710464616a - 615199049648140942921559884880 x^{9} + -446072941743783673296432194192a - 38150743447918437672254320048 x^{8} + 201360551746698280304024301520a - 189524272777557075741093867944 x^{7} + -265677565522723227247698452452a - 118642163062368886447075578624 x^{6} + (254676797500922063459334073296a + 600830195192731403460188475336 )x^{5} + (-119013055991881442440123830568a + 68072927592707452440962505984 )x^{4} + (321336035743965732955577865376a + 3533571039182922772059667488 )x^{3} + -199836185534017912835225620304a - 225771504676268458282285981632 x^{2} + (-426106624150245831086670588056a + 83850195656973474993212825632 )x + 479257542851214098174447001376a - 355244116824163404975967726562 \)
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