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ex.24.10.1.33_67_101.d

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 \(x^{2} + 90546471444873528678335943241b^{3} x + (-90546471444873528678335943241a\cdot b + 90546471444873528678335943241)b \)

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^{ 2 } \\ \chi^A\left(((2a\cdot \mu_3 + 2)b^{2} + 2\mu_3b + 4)c + (2a\cdot \mu_3 + (2a - 2))b^{2} + ((2a - 3)\mu_3 + (2a - 3))b + 4\mu_3 + 3 \right) &= i^{ 0 } \\ \chi^A\left(((2a + 4)\mu_3b^{2} + (4a + 2)\mu_3)c + 4\mu_3b^{2} + 4a\cdot \mu_3 + 1 \right) &= i^{ 2 } \\ \chi^A\left((((2a - 3)\mu_3 + (3a - 1))b^{2} + (2\mu_3 + (2a + 1))b + ((3a + 2)\mu_3 - 3a + 2))c + ((3a - 2)\mu_3 + (a + 4))b^{2} + ((2a + 3)\mu_3 + (a + 1))b + (2a + 4)\mu_3 - 2a + 1 \right) &= i^{ 0 } \\ \chi^A\left(((2a + 1)b^{2} + (2a + 2)b)\cdot c + (2a + 4)b^{2} + 2a\cdot b + 1 \right) &= i^{ 0 } \\ \chi^A\left((((3a + 2)\mu_3 + (2a + 3))b^{2} + (a + 1)\mu_3b + ((2a - 2)\mu_3 - 2))c + (3\mu_3 + (a - 2))b^{2} + (3\mu_3 + (a + 3))b + (2a - 2)\mu_3 - 3a + 1 \right) &= i^{ 0 } \\ \chi^A\left((((2a + 2)\mu_3 + 2)b^{2} + (a\cdot \mu_3 + a)b + ((-2a + 4)\mu_3 - a + 4))c + (3a\cdot \mu_3 + (a - 2))b^{2} + (-\mu_3 - 1)b + (-2a - 2)\mu_3 - 2a + 1 \right) &= i^{ 2 } \\ \chi^A\left(((4\mu_3 + 4)b^{2} + ((a - 2)\mu_3 + (2a - 2))b + ((4a - 2)\mu_3 + (4a - 2)))c + ((3a + 4)\mu_3 + (3a + 4))b^{2} + (\mu_3 + (2a - 3))b - 2\mu_3 - 1 \right) &= i^{ 0 } \\ \chi^A\left((((2a + 2)\mu_3 + (2a + 4))b^{2} + ((2a - 2)\mu_3 + (2a + 2))b + (4\mu_3 - 2a - 2))c + ((3a + 1)\mu_3 + 2)b^{2} + ((3a + 1)\mu_3 + (3a + 1))b + 4\mu_3 + a + 1 \right) &= i^{ 0 } \\ \chi^A\left(((-2\mu_3 + 4)b^{2} + ((3a + 2)\mu_3 + 4)b + (-3a - 2)\mu_3)c + a\cdot \mu_3b^{2} + (3a - 1)\mu_3b + (-a + 3)\mu_3 + 4a \right) &= i^{ 0 } \\ \chi^A\left(((4\mu_3 + 2)b^{2} + ((3a + 4)\mu_3 + (3a + 4))b + ((4a + 4)\mu_3 + (a - 2)))c + ((2a + 1)\mu_3 + (3a + 2))b^{2} + ((3a + 3)\mu_3 + (3a + 3))b - 2\mu_3 + 3a - 1 \right) &= i^{ 0 } \\ \chi^A\left((-\mu_3 + 3)c + -\mu_3 + 3 \right) &= i^{ 0 } \\ \chi^A\left((((3a + 2)\mu_3 + 2a)\cdot b^{2} + ((a + 4)\mu_3 + (3a - 2))b + (-3a - 2))c + ((2a - 1)\mu_3 + (3a + 2))b^{2} + ((a + 3)\mu_3 + (3a + 3))b + (-2a - 2)\mu_3 + a - 1 \right) &= i^{ 2 } \\ \chi^A\left(((2a + 1)\mu_3b^{2} + (2a + 2)\mu_3b)\cdot c + (2a + 4)\mu_3b^{2} + 2a\cdot \mu_3b + 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} + (-468574117940441383158947398788a + 348294242461322892559358706168 )x^{47} + -532197640475344068204458248944a - 69684520588617593052234005836 x^{46} + (399921728897077675123311701232a + 177076382925070468457917435360 )x^{45} + -79338723451210436172861548736a - 602365752224033155534140801276 x^{44} + (333722022334191334309814596764a + 153631105887000218799316606388 )x^{43} + 378658536280522094034240733108a - 48328937451728350599125836840 x^{42} + 295118401661102215574208003500a - 223756497528872617758594466504 x^{41} + -261241781812170281664431673056a - 491002175709539800408781702944 x^{40} + (596600855191465321504415678696a + 65167800083876961973067684440 )x^{39} + (234864916937254128550407483428a + 344679483499567227099852884276 )x^{38} + 24334959644229418117079367536a - 510279795235195526775686726436 x^{37} + -559728379156441230874645204376a - 47456773322492208315401111184 x^{36} + -400839755830538776608321840360a - 363385800235973698066206380800 x^{35} + -295829988209808100269412319580a - 595422028040368826273385632392 x^{34} + 267142210647423633737197869300a - 163375342669043866007436900568 x^{33} + (73665863451706030134588907560a + 354592087219867256043791628264 )x^{32} + 496176077681780089628781809576a - 254456844802178524514479690336 x^{31} + 567164484818770930860276965708a - 261183981648704387421281957716 x^{30} + -563038387605336791191490532276a - 360790929653748315402315776536 x^{29} + (461823235129718147661315338808a + 418911589365612591384518163352 )x^{28} + 250162750972388055225337895048a - 86911267083988281128004419896 x^{27} + (-394170220114745384554002265776a + 228912329357176391659413606676 )x^{26} + (605095821313843412077336411336a + 202381144759587832193320409056 )x^{25} + -533538473434909524366021505118a - 136476815315043864958990632460 x^{24} + 24760274616540795004516945512a - 468026476023106614046082622920 x^{23} + (290971569149141966692348501988a + 247740280703263508526717146616 )x^{22} + -592020201011961007519726089776a - 112456958640216950071498820752 x^{21} + 155583250323383934646227948476a - 376138258852086731429748549520 x^{20} + (-418572085767976640233513115372a + 69652245471288656648156313480 )x^{19} + (468743141508120258589063409440a + 549635079881809342984276219496 )x^{18} + (-584603752505709870516506622616a + 559993207060763110004861348024 )x^{17} + 279011092469214739922947330136a - 420934289793158581270909040672 x^{16} + -316102234941390956408400860336a - 54114955364795283458833568400 x^{15} + (181760550881577165418674273548a + 230104808471250239225107533568 )x^{14} + (332198041027200557478843097004a + 305960743219644596806341796040 )x^{13} + -181802824063140008065818107192a - 397099415316179520130339477428 x^{12} + -344440277402855063560419072120a - 320152081765207235993130108064 x^{11} + -524097983536891411516819402800a - 379377416863022498908290704168 x^{10} + 456021566478213303608367744704a - 289390433577844947945054611704 x^{9} + (312737372505155507816575286176a + 514987870993912333654394974200 )x^{8} + (196939448324732033982149726832a + 618186029822501770112206772864 )x^{7} + -599150490354295545655931831524a - 587802793877952636298324180400 x^{6} + -478802076828048401400982497648a - 38561554595350976425774290120 x^{5} + 389891236836248137652576605600a - 215033591667964540325261985600 x^{4} + (-228709408537475881662914616024a + 506926645252577542009619938624 )x^{3} + 18393751611192038875757345900a - 403306134735077773238187234696 x^{2} + 376312727493654112282897162120a - 528316368932778149777217050528 x - 451974891937317620051797668524a + 7791344630790621688715574790 \)
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