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Level	Weight	Analytic conductor	Analytic rank	Dim.

Bad $p$	Char.	Primitive character	Character order	Is maximal/largest

Coefficient field	Self-twists	CM/RM discriminant	Inner twist count	Is self-dual

Coefficient ring index	Coefficient ring gens.	Is twist minimal	Projective image

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Results (22 matches)

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Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				A-L signs		Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$	13	31	Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
5239.2.a.a	$5239$	$2$	5239.a	1.a	$1$	$1$	$1$	$41.834$	$\Q$	None	✓				403.2.f.a	$1$	$0$	$-1$	$0$	$3$	$-2$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-q^{4}+3q^{5}-2q^{7}+3q^{8}-3q^{9}+\cdots$
5239.2.a.b	$5239$	$2$	5239.a	1.a	$1$	$1$	$1$	$41.834$	$\Q$	None	✓				403.2.c.a	$1$	$1$	$0$	$2$	$-4$	$-2$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+2q^{3}-2q^{4}-4q^{5}-2q^{7}+q^{9}+\cdots$
5239.2.a.c	$5239$	$2$	5239.a	1.a	$1$	$1$	$1$	$41.834$	$\Q$	None	✓				403.2.c.a	$1$	$0$	$0$	$2$	$4$	$2$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+2q^{3}-2q^{4}+4q^{5}+2q^{7}+q^{9}+\cdots$
5239.2.a.d	$5239$	$2$	5239.a	1.a	$1$	$1$	$1$	$41.834$	$\Q$	None	✓				403.2.f.a	$1$	$1$	$1$	$0$	$-3$	$2$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-q^{4}-3q^{5}+2q^{7}-3q^{8}-3q^{9}+\cdots$
5239.2.a.e	$5239$	$2$	5239.a	1.a	$1$	$2$	$2$	$41.834$	$\Q(\sqrt{5}) $	None	✓				403.2.a.a	$1$	$0$	$-3$	$-4$	$0$	$-2$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+(-1-\beta )q^{2}-2q^{3}+3\beta q^{4}+(1-2\beta )q^{5}+\cdots$
5239.2.a.f	$5239$	$2$	5239.a	1.a	$1$	$2$	$2$	$41.834$	$\Q(\sqrt{5}) $	None	✓				31.2.a.a	$1$	$1$	$-1$	$-2$	$-2$	$4$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-\beta q^{2}-2\beta q^{3}+(-1+\beta )q^{4}-q^{5}+\cdots$
5239.2.a.g	$5239$	$2$	5239.a	1.a	$1$	$6$	$6$	$41.834$	6.6.5748973.1	None	✓				403.2.a.b	$1$	$1$	$2$	$-5$	$9$	$0$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-\beta _{4}q^{2}+(-1+\beta _{1})q^{3}+(1-\beta _{1}-\beta _{2}+\cdots)q^{4}+\cdots$
5239.2.a.h	$5239$	$2$	5239.a	1.a	$1$	$7$	$7$	$41.834$	$\mathbb{Q}[x]/(x^{7} - \cdots)$	None	✓				403.2.a.c	$1$	$1$	$-2$	$5$	$-11$	$-4$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-\beta _{3}q^{2}+(1-\beta _{1})q^{3}+(\beta _{1}+\beta _{4}+\beta _{6})q^{4}+\cdots$
5239.2.a.i	$5239$	$2$	5239.a	1.a	$1$	$8$	$8$	$41.834$	$\mathbb{Q}[x]/(x^{8} - \cdots)$	None	✓				403.2.a.e	$1$	$0$	$1$	$7$	$-11$	$2$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{2}+(1+\beta _{4})q^{3}+(1+\beta _{2})q^{4}+\cdots$
5239.2.a.j	$5239$	$2$	5239.a	1.a	$1$	$8$	$8$	$41.834$	$\mathbb{Q}[x]/(x^{8} - \cdots)$	None	✓				403.2.a.d	$1$	$0$	$5$	$-3$	$15$	$4$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+(1-\beta _{1})q^{2}-\beta _{6}q^{3}+(2-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots$
5239.2.a.k	$5239$	$2$	5239.a	1.a	$1$	$16$	$16$	$41.834$	$\mathbb{Q}[x]/(x^{16} - \cdots)$	None	✓				403.2.c.b	$1$	$1$	$-4$	$-2$	$-4$	$-2$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-\beta _{1}q^{2}-\beta _{6}q^{3}+(1+\beta _{2})q^{4}+\beta _{9}q^{5}+\cdots$
5239.2.a.l	$5239$	$2$	5239.a	1.a	$1$	$16$	$16$	$41.834$	$\mathbb{Q}[x]/(x^{16} - \cdots)$	None	✓				403.2.c.b	$1$	$0$	$4$	$-2$	$4$	$2$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{2}-\beta _{6}q^{3}+(1+\beta _{2})q^{4}-\beta _{9}q^{5}+\cdots$
5239.2.a.m	$5239$	$2$	5239.a	1.a	$1$	$17$	$17$	$41.834$	$\mathbb{Q}[x]/(x^{17} - \cdots)$	None	✓				403.2.f.b	$1$	$1$	$-4$	$0$	$-7$	$-6$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-\beta _{1}q^{2}+\beta _{9}q^{3}+(1+\beta _{2})q^{4}-\beta _{16}q^{5}+\cdots$
5239.2.a.n	$5239$	$2$	5239.a	1.a	$1$	$17$	$17$	$41.834$	$\mathbb{Q}[x]/(x^{17} - \cdots)$	None	✓				403.2.f.b	$1$	$0$	$4$	$0$	$7$	$6$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{2}+\beta _{9}q^{3}+(1+\beta _{2})q^{4}+\beta _{16}q^{5}+\cdots$
5239.2.a.o	$5239$	$2$	5239.a	1.a	$1$	$18$	$18$	$41.834$	$\mathbb{Q}[x]/(x^{18} - \cdots)$	None	✓				403.2.f.c	$1$	$1$	$-5$	$0$	$-6$	$-4$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-\beta _{1}q^{2}+\beta _{11}q^{3}+(1+\beta _{2})q^{4}+\beta _{7}q^{5}+\cdots$
5239.2.a.p	$5239$	$2$	5239.a	1.a	$1$	$18$	$18$	$41.834$	$\mathbb{Q}[x]/(x^{18} - \cdots)$	None	✓				403.2.f.c	$1$	$0$	$5$	$0$	$6$	$4$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{2}+\beta _{11}q^{3}+(1+\beta _{2})q^{4}-\beta _{7}q^{5}+\cdots$
5239.2.a.q	$5239$	$2$	5239.a	1.a	$1$	$34$	$34$	$41.834$		None	✓				403.2.r.a	$1$	$1$	$-8$	$0$	$-16$	$-8$	$-$	$-$	$+$		$\mathrm{SU}(2)$
5239.2.a.r	$5239$	$2$	5239.a	1.a	$1$	$34$	$34$	$41.834$		None	✓				403.2.r.a	$1$	$0$	$8$	$0$	$16$	$8$	$-$	$+$	$-$		$\mathrm{SU}(2)$
5239.2.a.s	$5239$	$2$	5239.a	1.a	$1$	$36$	$36$	$41.834$		None	✓	✓	✓		5239.2.a.s	$1$	$1$	$-2$	$-5$	$-5$	$-5$	$-$	$-$	$+$		$\mathrm{SU}(2)$
5239.2.a.t	$5239$	$2$	5239.a	1.a	$1$	$36$	$36$	$41.834$		None	✓	✓	✓		5239.2.a.s	$1$	$1$	$2$	$-5$	$5$	$5$	$+$	$+$	$+$		$\mathrm{SU}(2)$
5239.2.a.u	$5239$	$2$	5239.a	1.a	$1$	$54$	$54$	$41.834$		None	✓	✓	✓		5239.2.a.u	$1$	$0$	$-2$	$7$	$-5$	$-5$	$-$	$+$	$-$		$\mathrm{SU}(2)$
5239.2.a.v	$5239$	$2$	5239.a	1.a	$1$	$54$	$54$	$41.834$		None	✓	✓	✓		5239.2.a.u	$1$	$0$	$2$	$7$	$5$	$5$	$+$	$-$	$-$		$\mathrm{SU}(2)$

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Elliptic curves over $\Q$
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This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.