ex.24.10.1.1_3_5.a
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 \(x^{2} + 253530120045645880299340641075b x - 63382530011411470074835160269b \)
Underlying Character
Character \(\chi^A:\mathcal O_K^\times \to \mathbb C^\times\) with the following properties:
Order
4
Conductor exponent
24
Values on generators of
\((\mathcal{O}_K/\mathfrak p^{ 24 })^\times/U_{\mathfrak{p}^{ 24 } }\)
:
\(\begin{array}{l}
\chi^A\left(\mu_3c + 1 \right) &= i^{ 1 }
\\
\chi^A\left((((-2a + 4)\mu_3 - 2a + 4)b^{2} + ((2a + 2)\mu_3 + 4a)\cdot b - 2\mu_3)c + ((a + 2)\mu_3 + a)b^{2} + ((4a - 2)\mu_3 + (2a - 2))b + (-2a - 2)\mu_3 + 2a + 1 \right) &= i^{ 2 }
\\
\chi^A\left(((-2\mu_3 + (4a - 2))b^{2} + (4\mu_3 + 4)b + ((2a + 4)\mu_3 + (2a + 4)))c + ((-2a + 2)\mu_3 + (4a + 2))b^{2} + ((2a + 3)\mu_3 + (2a + 3))b + (4a + 1)\mu_3 + 2a + 1 \right) &= i^{ 0 }
\\
\chi^A\left((((4a - 1)\mu_3 - 1)b^{2} + (-2\mu_3 + (3a + 4))b + ((-a - 2)\mu_3 + (a - 2)))c + ((2a - 2)\mu_3 + (2a + 1))b^{2} + ((-a + 3)\mu_3 + (2a - 1))b + (2a - 2)\mu_3 + a - 3 \right) &= i^{ 0 }
\\
\chi^A\left(((\mu_3 + (3a + 3))b^{2} + ((2a - 2)\mu_3 + (3a + 2))b + ((3a + 2)\mu_3 - a))c + ((-3a - 2)\mu_3 - 3a + 2)b^{2} + (4a\cdot \mu_3 - 2a - 3)b + (-a + 1)\mu_3 - 2a - 3 \right) &= i^{ 0 }
\\
\chi^A\left((4b^{2} - \mu_3 + 3)c + 4b^{2} - \mu_3 + 4a + 3 \right) &= i^{ 0 }
\\
\chi^A\left(((3a + 2)\mu_3b^{2} + (-2a - 2)\mu_3b + (2a - 2)\mu_3)c + (-a + 3)\mu_3b^{2} + (3a + 1)\mu_3b + (a + 1)\mu_3 \right) &= i^{ 0 }
\\
\chi^A\left(((2\mu_3 + (4a - 2))b^{2} - 2a\cdot b + 2a\cdot \mu_3 - 2)c + ((-2a - 2)\mu_3 + (4a + 4))b^{2} + ((2a - 1)\mu_3 - 2a + 3)b + (4a + 1)\mu_3 + 1 \right) &= i^{ 2 }
\\
\chi^A\left((((3a + 2)\mu_3 + (3a + 2))b^{2} + ((2a + 4)\mu_3 + (2a + 4))b + (4a\cdot \mu_3 + 4a))\cdot c + ((4a + 4)\mu_3 + (4a + 4))b^{2} + ((2a - 1)\mu_3 + (2a - 1))b + \mu_3 + 1 \right) &= i^{ 0 }
\\
\chi^A\left(((4a + 4)\mu_3b^{2} + (4a\cdot \mu_3 + 4a)\cdot b + 4a)\cdot c + ((-a + 1)\mu_3 + (a + 3))b^{2} + ((3a - 3)\mu_3 - 3a - 1)b + (4a + 4)\mu_3 - 1 \right) &= i^{ 2 }
\\
\chi^A\left((((4a + 2)\mu_3 + 4a)\cdot b^{2} + ((4a - 2)\mu_3 - 2a + 4)b + ((4a + 4)\mu_3 + (4a + 2)))c + (-2a - 1)b^{2} + ((a + 1)\mu_3 + (a + 1))b - 2\mu_3 + a - 3 \right) &= i^{ 2 }
\\
\chi^A\left((((-a - 2)\mu_3 + 2)b^{2} + ((4a - 2)\mu_3 + 2a)\cdot b + ((a + 2)\mu_3 + a))c + (-3a\cdot \mu_3 - 2a + 3)b^{2} + ((2a + 1)\mu_3 - 3)b + (4a + 4)\mu_3 + 3a + 3 \right) &= i^{ 0 }
\\
\chi^A\left((((2a + 4)\mu_3 + (2a + 4))b^{2} + ((2a - 2)\mu_3 + (2a + 2))b + ((-2a + 4)\mu_3 + (4a - 2)))c + ((-a + 2)\mu_3 + (2a - 3))b^{2} + ((4a + 4)\mu_3 + (3a - 3))b + \mu_3 + 3 \right) &= i^{ 2 }
\\
\chi^A\left(((-2a - 2)b^{2} + (-a + 2)b + 2)c + (4a - 2)b^{2} + 4b - 3 \right) &= i^{ 0 }
\end{array}
\)
Inertia Polynomial
The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + 28 x^{47} + (8a + 20 )x^{46} + (14a + 20 )x^{45} + (14a + 8 )x^{44} + (28a + 12 )x^{43} + (7a + 18 )x^{42} + (14a + 12 )x^{41} + (24a + 6 )x^{40} + (26a + 12 )x^{39} + (12a + 8 )x^{38} + (30a + 4 )x^{37} + (16a + 6 )x^{36} + 28 x^{35} + (14a + 22 )x^{34} + 18a x^{33} + (6a + 26 )x^{32} + (4a + 8 )x^{31} + (18a + 14 )x^{30} + 24 x^{29} + (26a + 28 )x^{28} + (28a + 4 )x^{27} + (28a + 26 )x^{26} + (16a + 28 )x^{25} + (10a + 2 )x^{24} + (20a + 20 )x^{23} + (28a + 4 )x^{22} + 8a x^{21} + (30a + 24 )x^{20} + (20a + 28 )x^{19} + (4a + 30 )x^{18} + (8a + 20 )x^{17} + (10a + 8 )x^{16} + (4a + 16 )x^{15} + (28a + 20 )x^{14} + (16a + 12 )x^{13} + (14a + 26 )x^{12} + (28a + 16 )x^{11} + (30a + 16 )x^{10} + 28 x^{9} + (28a + 4 )x^{8} + (20a + 16 )x^{7} + (18a + 8 )x^{6} + (12a + 8 )x^{5} + 24 x^{3} + (30a + 20 )x^{2} + 30a + 2 \)