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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 (41 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*	$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$	3	11	23	Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
8349.2.a.a	$8349$	$2$	8349.a	1.a	$1$	$1$	$1$	$66.667$	$\Q$	None	✓				69.2.a.a	$1$	$1$	$-1$	$1$	$0$	$2$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{3}-q^{4}-q^{6}+2q^{7}+3q^{8}+\cdots$
8349.2.a.b	$8349$	$2$	8349.a	1.a	$1$	$1$	$1$	$66.667$	$\Q$	None	✓	✓			8349.2.a.b	$1$	$1$	$-1$	$1$	$1$	$0$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{3}-q^{4}+q^{5}-q^{6}+3q^{8}+\cdots$
8349.2.a.c	$8349$	$2$	8349.a	1.a	$1$	$1$	$1$	$66.667$	$\Q$	None	✓				759.2.a.a	$1$	$1$	$1$	$-1$	$0$	$2$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-q^{3}-q^{4}-q^{6}+2q^{7}-3q^{8}+\cdots$
8349.2.a.d	$8349$	$2$	8349.a	1.a	$1$	$1$	$1$	$66.667$	$\Q$	None	✓				759.2.a.b	$1$	$1$	$1$	$1$	$-2$	$0$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{3}-q^{4}-2q^{5}+q^{6}-3q^{8}+\cdots$
8349.2.a.e	$8349$	$2$	8349.a	1.a	$1$	$1$	$1$	$66.667$	$\Q$	None	✓	✓			8349.2.a.b	$1$	$1$	$1$	$1$	$1$	$0$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{3}-q^{4}+q^{5}+q^{6}-3q^{8}+\cdots$
8349.2.a.f	$8349$	$2$	8349.a	1.a	$1$	$2$	$2$	$66.667$	$\Q(\sqrt{2}) $	None	✓				759.2.a.d	$1$	$0$	$-2$	$2$	$4$	$-4$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+(-1+\beta )q^{2}+q^{3}+(1-2\beta )q^{4}+2q^{5}+\cdots$
8349.2.a.g	$8349$	$2$	8349.a	1.a	$1$	$2$	$2$	$66.667$	$\Q(\sqrt{3}) $	None	✓	✓			8349.2.a.g	$2$	$0$	$0$	$-2$	$-6$	$0$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+\beta q^{2}-q^{3}+q^{4}-3q^{5}-\beta q^{6}+3\beta q^{7}+\cdots$
8349.2.a.h	$8349$	$2$	8349.a	1.a	$1$	$2$	$2$	$66.667$	$\Q(\sqrt{3}) $	None	✓	✓			8349.2.a.h	$2$	$1$	$0$	$-2$	$0$	$0$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+\beta q^{2}-q^{3}+q^{4}-\beta q^{6}+2\beta q^{7}+\cdots$
8349.2.a.i	$8349$	$2$	8349.a	1.a	$1$	$2$	$2$	$66.667$	$\Q(\sqrt{5}) $	None	✓				69.2.a.b	$1$	$1$	$0$	$-2$	$-2$	$-2$	$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	$q-\beta q^{2}-q^{3}+3q^{4}+(-1-\beta )q^{5}+\cdots$
8349.2.a.j	$8349$	$2$	8349.a	1.a	$1$	$2$	$2$	$66.667$	$\Q(\sqrt{5}) $	None	✓				759.2.a.c	$1$	$0$	$1$	$2$	$-2$	$-4$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+\beta q^{2}+q^{3}+(-1+\beta )q^{4}-q^{5}+\beta q^{6}+\cdots$
8349.2.a.k	$8349$	$2$	8349.a	1.a	$1$	$3$	$3$	$66.667$	3.3.148.1	None	✓				759.2.a.e	$1$	$0$	$1$	$-3$	$-4$	$0$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{2}-q^{3}+(\beta _{1}+\beta _{2})q^{4}+(-1+\cdots)q^{5}+\cdots$
8349.2.a.l	$8349$	$2$	8349.a	1.a	$1$	$3$	$3$	$66.667$	3.3.148.1	None	✓				759.2.a.f	$1$	$0$	$1$	$3$	$-2$	$6$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{2}+q^{3}+(\beta _{1}+\beta _{2})q^{4}+(-1+\cdots)q^{5}+\cdots$
8349.2.a.m	$8349$	$2$	8349.a	1.a	$1$	$5$	$5$	$66.667$	5.5.1563364.1	None	✓				759.2.a.g	$1$	$0$	$2$	$5$	$4$	$0$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{2}+q^{3}+(2+\beta _{2})q^{4}+(1+\beta _{1}+\cdots)q^{5}+\cdots$
8349.2.a.n	$8349$	$2$	8349.a	1.a	$1$	$6$	$6$	$66.667$	6.6.4222000.1	None	✓				759.2.a.h	$1$	$1$	$-2$	$-6$	$6$	$0$	$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	$q-\beta _{1}q^{2}-q^{3}+(\beta _{1}+\beta _{2})q^{4}+(1-\beta _{3}+\cdots)q^{5}+\cdots$
8349.2.a.o	$8349$	$2$	8349.a	1.a	$1$	$6$	$6$	$66.667$	6.6.199374400.1	None	✓	✓			8349.2.a.o	$2$	$1$	$0$	$-6$	$0$	$0$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}+\beta _{3}q^{5}+\cdots$
8349.2.a.p	$8349$	$2$	8349.a	1.a	$1$	$6$	$6$	$66.667$	6.6.3356224.1	None	✓	✓			8349.2.a.p	$2$	$1$	$0$	$6$	$-8$	$0$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{2}+q^{3}+\beta _{2}q^{4}+(-1-\beta _{2}+\cdots)q^{5}+\cdots$
8349.2.a.q	$8349$	$2$	8349.a	1.a	$1$	$7$	$7$	$66.667$	$\mathbb{Q}[x]/(x^{7} - \cdots)$	None	✓	✓			8349.2.a.q	$1$	$1$	$-2$	$7$	$-9$	$-1$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-\beta _{3}q^{2}+q^{3}+(1-\beta _{4}-\beta _{6})q^{4}+(-2+\cdots)q^{5}+\cdots$
8349.2.a.r	$8349$	$2$	8349.a	1.a	$1$	$7$	$7$	$66.667$	$\mathbb{Q}[x]/(x^{7} - \cdots)$	None	✓	✓			8349.2.a.r	$1$	$1$	$-1$	$-7$	$0$	$2$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-\beta _{1}q^{2}-q^{3}+\beta _{2}q^{4}-\beta _{5}q^{5}+\beta _{1}q^{6}+\cdots$
8349.2.a.s	$8349$	$2$	8349.a	1.a	$1$	$7$	$7$	$66.667$	$\mathbb{Q}[x]/(x^{7} - \cdots)$	None	✓	✓			8349.2.a.r	$1$	$1$	$1$	$-7$	$0$	$-2$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{2}-q^{3}+\beta _{2}q^{4}-\beta _{5}q^{5}-\beta _{1}q^{6}+\cdots$
8349.2.a.t	$8349$	$2$	8349.a	1.a	$1$	$7$	$7$	$66.667$	$\mathbb{Q}[x]/(x^{7} - \cdots)$	None	✓	✓			8349.2.a.q	$1$	$1$	$2$	$7$	$-9$	$1$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+\beta _{3}q^{2}+q^{3}+(1-\beta _{4}-\beta _{6})q^{4}+(-2+\cdots)q^{5}+\cdots$
8349.2.a.u	$8349$	$2$	8349.a	1.a	$1$	$8$	$8$	$66.667$	$\mathbb{Q}[x]/(x^{8} - \cdots)$	None	✓				759.2.a.i	$1$	$0$	$-2$	$-8$	$6$	$6$	$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	$q-\beta _{1}q^{2}-q^{3}+(2+\beta _{2})q^{4}+(1+\beta _{4}+\cdots)q^{5}+\cdots$
8349.2.a.v	$8349$	$2$	8349.a	1.a	$1$	$8$	$8$	$66.667$	$\mathbb{Q}[x]/(x^{8} - \cdots)$	None	✓				759.2.a.j	$1$	$1$	$-2$	$8$	$0$	$-6$	$-$	$-$	$+$	$+$	$2^{4}$	$\mathrm{SU}(2)$	$q-\beta _{1}q^{2}+q^{3}+(1+\beta _{2})q^{4}-\beta _{6}q^{5}+\cdots$
8349.2.a.w	$8349$	$2$	8349.a	1.a	$1$	$8$	$8$	$66.667$	8.8.$\cdots$.1	None	✓	✓			8349.2.a.w	$2$	$0$	$0$	$-8$	$-10$	$0$	$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}+(-1+\beta _{5}+\cdots)q^{5}+\cdots$
8349.2.a.x	$8349$	$2$	8349.a	1.a	$1$	$10$	$10$	$66.667$	$\mathbb{Q}[x]/(x^{10} - \cdots)$	None	✓	✓			8349.2.a.x	$2$	$0$	$0$	$10$	$4$	$0$	$-$	$+$	$+$	$-$	$2\cdot 5$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{2}+q^{3}+(1+\beta _{2})q^{4}-\beta _{7}q^{5}+\cdots$
8349.2.a.y	$8349$	$2$	8349.a	1.a	$1$	$11$	$11$	$66.667$	$\mathbb{Q}[x]/(x^{11} - \cdots)$	None	✓	✓			8349.2.a.y	$1$	$0$	$-1$	$-11$	$-4$	$0$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}+\beta _{5}q^{5}+\cdots$
8349.2.a.z	$8349$	$2$	8349.a	1.a	$1$	$11$	$11$	$66.667$	$\mathbb{Q}[x]/(x^{11} - \cdots)$	None	✓	✓			8349.2.a.y	$1$	$0$	$1$	$-11$	$-4$	$0$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}+\beta _{5}q^{5}+\cdots$
8349.2.a.ba	$8349$	$2$	8349.a	1.a	$1$	$12$	$12$	$66.667$	$\mathbb{Q}[x]/(x^{12} - \cdots)$	None	✓	✓			8349.2.a.ba	$1$	$0$	$-1$	$12$	$4$	$1$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-\beta _{1}q^{2}+q^{3}+(1+\beta _{2})q^{4}-\beta _{4}q^{5}+\cdots$
8349.2.a.bb	$8349$	$2$	8349.a	1.a	$1$	$12$	$12$	$66.667$	$\mathbb{Q}[x]/(x^{12} - \cdots)$	None	✓	✓			8349.2.a.bb	$2$	$0$	$0$	$-12$	$14$	$0$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+(-\beta _{1}+\beta _{6})q^{2}-q^{3}+(1+\beta _{3}+\beta _{7}+\cdots)q^{4}+\cdots$
8349.2.a.bc	$8349$	$2$	8349.a	1.a	$1$	$12$	$12$	$66.667$	$\mathbb{Q}[x]/(x^{12} - \cdots)$	None	✓	✓			8349.2.a.bc	$2$	$0$	$0$	$-12$	$-4$	$0$	$+$	$+$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{2}-q^{3}+(2+\beta _{2})q^{4}+\beta _{4}q^{5}+\cdots$
8349.2.a.bd	$8349$	$2$	8349.a	1.a	$1$	$12$	$12$	$66.667$	$\mathbb{Q}[x]/(x^{12} - \cdots)$	None	✓	✓			8349.2.a.bd	$2$	$1$	$0$	$12$	$-10$	$0$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{2}+q^{3}+\beta _{2}q^{4}+(-1-\beta _{6}+\cdots)q^{5}+\cdots$
8349.2.a.be	$8349$	$2$	8349.a	1.a	$1$	$12$	$12$	$66.667$	$\mathbb{Q}[x]/(x^{12} - \cdots)$	None	✓	✓			8349.2.a.ba	$1$	$0$	$1$	$12$	$4$	$-1$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{2}+q^{3}+(1+\beta _{2})q^{4}-\beta _{4}q^{5}+\cdots$
8349.2.a.bf	$8349$	$2$	8349.a	1.a	$1$	$14$	$14$	$66.667$	$\mathbb{Q}[x]/(x^{14} - \cdots)$	None	✓	✓			8349.2.a.bf	$2$	$1$	$0$	$-14$	$6$	$0$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{2}-q^{3}+(\beta _{5}-\beta _{6})q^{4}-\beta _{6}q^{5}+\cdots$
8349.2.a.bg	$8349$	$2$	8349.a	1.a	$1$	$20$	$20$	$66.667$	$\mathbb{Q}[x]/(x^{20} - \cdots)$	None	✓	✓			8349.2.a.bg	$2$	$0$	$0$	$20$	$14$	$0$	$-$	$+$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{2}+q^{3}+(1+\beta _{2})q^{4}+(1-\beta _{9}+\cdots)q^{5}+\cdots$
8349.2.a.bh	$8349$	$2$	8349.a	1.a	$1$	$22$	$22$	$66.667$		None	✓		✓		759.2.i.d	$1$	$1$	$-10$	$22$	$1$	$-7$	$-$	$+$	$-$	$+$		$\mathrm{SU}(2)$
8349.2.a.bi	$8349$	$2$	8349.a	1.a	$1$	$22$	$22$	$66.667$		None	✓		✓		759.2.i.a	$1$	$1$	$-6$	$22$	$1$	$-9$	$-$	$-$	$+$	$+$		$\mathrm{SU}(2)$
8349.2.a.bj	$8349$	$2$	8349.a	1.a	$1$	$22$	$22$	$66.667$		None	✓		✓		759.2.i.b	$1$	$1$	$-2$	$-22$	$-1$	$-9$	$+$	$-$	$-$	$+$		$\mathrm{SU}(2)$
8349.2.a.bk	$8349$	$2$	8349.a	1.a	$1$	$22$	$22$	$66.667$		None	✓		✓		759.2.i.c	$1$	$0$	$-2$	$-22$	$-1$	$7$	$+$	$-$	$+$	$-$		$\mathrm{SU}(2)$
8349.2.a.bl	$8349$	$2$	8349.a	1.a	$1$	$22$	$22$	$66.667$		None	✓		✓		759.2.i.c	$1$	$1$	$2$	$-22$	$-1$	$-7$	$+$	$+$	$+$	$+$		$\mathrm{SU}(2)$
8349.2.a.bm	$8349$	$2$	8349.a	1.a	$1$	$22$	$22$	$66.667$		None	✓		✓		759.2.i.b	$1$	$0$	$2$	$-22$	$-1$	$9$	$+$	$+$	$-$	$-$		$\mathrm{SU}(2)$
8349.2.a.bn	$8349$	$2$	8349.a	1.a	$1$	$22$	$22$	$66.667$		None	✓		✓		759.2.i.a	$1$	$0$	$6$	$22$	$1$	$9$	$-$	$+$	$+$	$-$		$\mathrm{SU}(2)$
8349.2.a.bo	$8349$	$2$	8349.a	1.a	$1$	$22$	$22$	$66.667$		None	✓		✓		759.2.i.d	$1$	$0$	$10$	$22$	$1$	$7$	$-$	$-$	$-$	$-$		$\mathrm{SU}(2)$

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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
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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.