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ex.24.10.1.131_259_387.d

Base Field
\(F = \) 2.1.2.3a1.2 \( = \mathbb{Q}_{ 2 }(a) = \mathbb{Q}_{ 2 }[x] / (x^{2} + 5\cdot 2 )\) 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 \(x^{2} + 253530120045645880299340641075b^{4} x + 3b \)

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

Inertia Polynomial

The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + (-117468031358374316604206598064a + 25747429534619484384242546584 )x^{47} + 481273540167406769634356066124a - 55503193956699812987712010736 x^{46} + -477858571428712840018027648220a - 398134175282120491634022614064 x^{45} + -186231176155076868319683439608a - 309075878469798129290418501500 x^{44} + (-158213631313717592297030995736a + 104516386406892251799565157680 )x^{43} + -612871123977962910157923267164a - 277212805515281044096502019828 x^{42} + (-498327082551280271845499284448a + 170978437314262061290002781264 )x^{41} + -331462544482140839444751729596a - 427320395608009214387106349380 x^{40} + 514619005742968427553230691096a - 256349573780042841271253598640 x^{39} + -445132710966478898395785000452a - 339884406928240100217064712040 x^{38} + 452915083147385384211560028260a - 63884383290390573341933147784 x^{37} + -256192867891620188624202456900a - 586885184776582674158684936852 x^{36} + -506519860702728672501995730672a - 34160731908081723486098189984 x^{35} + 237073968983576587644464680484a - 517835322959704505789388249904 x^{34} + (465890543382818819142615475736a + 616674842440909596904431260064 )x^{33} + -440818351688987985680387679440a - 159490137782039675358518560256 x^{32} + 499580262006730380793799625344a - 440876271466783702895165454144 x^{31} + (466460810571742956641426888224a + 446387759083205805257085569376 )x^{30} + 449990456640642715910363042544a - 51226179138803729421639989448 x^{29} + (-541510649185914134730967956808a + 587130210191958424663059634344 )x^{28} + 219529533282881128557372795760a - 46199687118368225179690073648 x^{27} + (-67396473106721417080749022804a + 529681969684509134021305938048 )x^{26} + (523967354017802093890077871248a + 477137467604052608284141274840 )x^{25} + (-473770196408603853773212553910a + 417465027100773066259478895836 )x^{24} + -165136646461620963153444814136a - 568797796791831417293214011024 x^{23} + -575922467800854264017225288416a - 578873708048096999587814315640 x^{22} + (-104370856150187456569245864344a + 283064459421493768237383878840 )x^{21} + (178015646102291809473060642372a + 215318164995758315957823594792 )x^{20} + (574959295499190438431294225968a + 399378366178644020141698324240 )x^{19} + -378093713271868943895837383892a - 226817068070596856699244581248 x^{18} + (76382662881757954781107130416a + 122868008915222197410521563664 )x^{17} + (248965611657067909166610044268a + 42787777746726102713305708896 )x^{16} + (-150099676749914760168057616528a + 26560388542907800313832552896 )x^{15} + (-281234614698682542077568883752a + 271426459013366136271990418440 )x^{14} + -487480878527466488196092848432a - 107556872100727279359585572472 x^{13} + 327633935075015118538324547104a - 369239366375033022535665440976 x^{12} + (116161608744858545724677365520a + 304322846190308343212968388384 )x^{11} + (604409966227216778203686128432a + 485497162314696380006695567816 )x^{10} + (531345419196400396418021025104a + 343700687251491554115802471152 )x^{9} + (383517928022604181978889378592a + 302012043735306095212993852976 )x^{8} + -275295522879612200267775505504a - 448635959576980864108435008368 x^{7} + (-122621611076478493220415843664a + 523426204873795427206582613344 )x^{6} + (-153397775059571699111903876264a + 593243607439059610945605784368 )x^{5} + (-467762510731525291120412405496a + 614099083976586548447699374912 )x^{4} + -26580959221304491864335763376a - 423822839450113108363154022624 x^{3} + -516229418033658198191151027944a - 269668577922521004347060349144 x^{2} + (-319149779358390569133764782136a + 566753806745605520168924303376 )x - 89531565552835244851716004940a + 305841607054889449095694345554 \)
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