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Level	Weight	Analytic conductor	$Nk^2$	Dim.

Bad $p$	Char.	Primitive character	Character order	Coefficient field

Self-twists	CM/RM discriminant	Inner twist count	Is self-dual	Analytic rank

Coefficient ring index	Coefficient ring gens.		Projective image	Projective image type
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Results (45 matches)

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Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces				Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$	Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
422.2.a.a	$422$	$2$	422.a	1.a	$1$	$1$	$1$	$3.370$	$\Q$	None	✓	$1$	$1$	$-1$	$0$	$1$	$-2$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}+q^{5}-2q^{7}-q^{8}-3q^{9}+\cdots$
422.2.a.b	$422$	$2$	422.a	1.a	$1$	$2$	$2$	$3.370$	$\Q(\sqrt{5}) $	None	✓	$1$	$0$	$-2$	$3$	$2$	$8$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+(1+\beta )q^{3}+q^{4}+(2-2\beta )q^{5}+\cdots$
422.2.a.c	$422$	$2$	422.a	1.a	$1$	$3$	$3$	$3.370$	3.3.257.1	None	✓	$1$	$1$	$-3$	$-1$	$-5$	$-5$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+(-\beta _{1}+\beta _{2})q^{3}+q^{4}+(-2+\cdots)q^{5}+\cdots$
422.2.a.d	$422$	$2$	422.a	1.a	$1$	$3$	$3$	$3.370$	3.3.785.1	None	✓	$1$	$0$	$-3$	$-1$	$1$	$1$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-\beta _{1}q^{3}+q^{4}+\beta _{2}q^{5}+\beta _{1}q^{6}+\cdots$
422.2.a.e	$422$	$2$	422.a	1.a	$1$	$3$	$3$	$3.370$	$\Q(\zeta_{14})^+$	None	✓	$1$	$1$	$3$	$-5$	$-3$	$-9$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+(-2-\beta _{2})q^{3}+q^{4}+(-1+\beta _{1}+\cdots)q^{5}+\cdots$
422.2.a.f	$422$	$2$	422.a	1.a	$1$	$6$	$6$	$3.370$	6.6.43983893.1	None	✓	$1$	$0$	$6$	$4$	$-2$	$7$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+(1-\beta _{1})q^{3}+q^{4}-\beta _{5}q^{5}+(1+\cdots)q^{6}+\cdots$
422.2.c.a	$422$	$2$	422.c	211.c	$3$	$4$	$2$	$3.370$	$\Q(\sqrt{-3}, \sqrt{29})$	None		$2$	$0$	$2$	$-2$	$6$	$0$		$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(1-\beta _{2})q^{2}+(-1+\beta _{2})q^{3}-\beta _{2}q^{4}+\cdots$
422.2.c.b	$422$	$2$	422.c	211.c	$3$	$4$	$2$	$3.370$	$\Q(\sqrt{-3}, \sqrt{5})$	None		$2$	$0$	$2$	$0$	$-2$	$4$		$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(1+\beta _{3})q^{2}+(-1+2\beta _{1}-\beta _{3})q^{3}+\cdots$
422.2.c.c	$422$	$2$	422.c	211.c	$3$	$8$	$4$	$3.370$	8.0.447703281.1	None		$2$	$0$	$4$	$2$	$-6$	$-7$		$3$	$\mathrm{SU}(2)[C_{3}]$	$q+(1-\beta _{2})q^{2}+(\beta _{3}-\beta _{4}+\beta _{7})q^{3}-\beta _{2}q^{4}+\cdots$
422.2.c.d	$422$	$2$	422.c	211.c	$3$	$18$	$9$	$3.370$	$\mathbb{Q}[x]/(x^{18} - \cdots)$	None		$2$	$0$	$-9$	$-2$	$2$	$-1$		$2^{4}$	$\mathrm{SU}(2)[C_{3}]$	$q+\beta _{5}q^{2}-\beta _{1}q^{3}+(-1-\beta _{5})q^{4}+\beta _{6}q^{5}+\cdots$
422.2.d.a	$422$	$2$	422.d	211.d	$5$	$16$	$4$	$3.370$	$\mathbb{Q}[x]/(x^{16} - \cdots)$	None		$2$	$0$	$4$	$2$	$8$	$0$		$5$	$\mathrm{SU}(2)[C_{5}]$	$q+\beta _{5}q^{2}+(-\beta _{4}-\beta _{5}+\beta _{7}+\beta _{8}+\beta _{11}+\cdots)q^{3}+\cdots$
422.2.d.b	$422$	$2$	422.d	211.d	$5$	$20$	$5$	$3.370$	$\mathbb{Q}[x]/(x^{20} - \cdots)$	None		$2$	$0$	$5$	$-3$	$-7$	$8$		$1$	$\mathrm{SU}(2)[C_{5}]$	$q-\beta _{10}q^{2}+\beta _{6}q^{3}+(-1-\beta _{9}-\beta _{10}+\cdots)q^{4}+\cdots$
422.2.d.c	$422$	$2$	422.d	211.d	$5$	$40$	$10$	$3.370$		None		$2$	$0$	$-10$	$2$	$-1$	$4$			$\mathrm{SU}(2)[C_{5}]$
422.2.f.a	$422$	$2$	422.f	211.f	$7$	$54$	$9$	$3.370$		None		$2$	$0$	$9$	$-1$	$1$	$-2$			$\mathrm{SU}(2)[C_{7}]$
422.2.f.b	$422$	$2$	422.f	211.f	$7$	$60$	$10$	$3.370$		None		$2$	$0$	$-10$	$-3$	$-1$	$-6$			$\mathrm{SU}(2)[C_{7}]$
422.2.i.a	$422$	$2$	422.i	211.i	$15$	$64$	$8$	$3.370$		None		$2$	$0$	$-8$	$0$	$2$	$-2$			$\mathrm{SU}(2)[C_{15}]$
422.2.i.b	$422$	$2$	422.i	211.i	$15$	$72$	$9$	$3.370$		None		$2$	$0$	$9$	$-3$	$-2$	$-4$			$\mathrm{SU}(2)[C_{15}]$
422.2.j.a	$422$	$2$	422.j	211.j	$21$	$96$	$8$	$3.370$		None		$2$	$0$	$-8$	$0$	$2$	$3$			$\mathrm{SU}(2)[C_{21}]$
422.2.j.b	$422$	$2$	422.j	211.j	$21$	$108$	$9$	$3.370$		None		$2$	$0$	$9$	$2$	$-2$	$1$			$\mathrm{SU}(2)[C_{21}]$
422.2.l.a	$422$	$2$	422.l	211.l	$35$	$216$	$9$	$3.370$		None		$2$	$0$	$-9$	$1$	$-1$	$-8$			$\mathrm{SU}(2)[C_{35}]$
422.2.l.b	$422$	$2$	422.l	211.l	$35$	$240$	$10$	$3.370$		None		$2$	$0$	$10$	$-2$	$1$	$-4$			$\mathrm{SU}(2)[C_{35}]$
422.2.o.a	$422$	$2$	422.o	211.o	$105$	$384$	$8$	$3.370$		None		$2$	$0$	$8$	$0$	$-2$	$2$			$\mathrm{SU}(2)[C_{105}]$
422.2.o.b	$422$	$2$	422.o	211.o	$105$	$432$	$9$	$3.370$		None		$2$	$0$	$-9$	$3$	$2$	$4$			$\mathrm{SU}(2)[C_{105}]$
422.3.b.a	$422$	$3$	422.b	211.b	$2$	$34$	$34$	$11.499$		None		$2$	$0$	$0$	$0$	$0$	$0$			$\mathrm{SU}(2)[C_{2}]$
422.3.e.a	$422$	$3$	422.e	211.e	$6$	$72$	$36$	$11.499$		None		$2$	$0$	$0$	$-6$	$0$	$18$			$\mathrm{SU}(2)[C_{6}]$
422.3.g.a	$422$	$3$	422.g	211.g	$10$	$136$	$34$	$11.499$		None		$2$	$0$	$0$	$-10$	$0$	$-10$			$\mathrm{SU}(2)[C_{10}]$
422.3.h.a	$422$	$3$	422.h	211.h	$14$	$204$	$34$	$11.499$		None		$2$	$0$	$0$	$0$	$0$	$0$			$\mathrm{SU}(2)[C_{14}]$
422.3.k.a	$422$	$3$	422.k	211.k	$30$	$288$	$36$	$11.499$		None		$2$	$0$	$0$	$16$	$0$	$42$			$\mathrm{SU}(2)[C_{30}]$
422.3.m.a	$422$	$3$	422.m	211.m	$42$	$432$	$36$	$11.499$		None		$2$	$0$	$0$	$6$	$0$	$-18$			$\mathrm{SU}(2)[C_{42}]$
422.3.n.a	$422$	$3$	422.n	211.n	$70$	$816$	$34$	$11.499$		None		$2$	$0$	$0$	$10$	$0$	$10$			$\mathrm{SU}(2)[C_{70}]$
422.3.p.a	$422$	$3$	422.p	211.p	$210$	$1728$	$36$	$11.499$		None		$2$	$0$	$0$	$-16$	$0$	$-42$			$\mathrm{SU}(2)[C_{210}]$
422.4.a.a	$422$	$4$	422.a	1.a	$1$	$3$	$3$	$24.899$	3.3.3873.1	None	✓	$1$	$1$	$6$	$-11$	$-4$	$-24$	$-$	$1$	$\mathrm{SU}(2)$	$q+2q^{2}+(-4+\beta _{1})q^{3}+4q^{4}+(-1+\cdots)q^{5}+\cdots$
422.4.a.b	$422$	$4$	422.a	1.a	$1$	$8$	$8$	$24.899$	$\mathbb{Q}[x]/(x^{8} - \cdots)$	None	✓	$1$	$1$	$16$	$-4$	$-16$	$-58$	$-$	$2^{4}$	$\mathrm{SU}(2)$	$q+2q^{2}+\beta _{3}q^{3}+4q^{4}+(-2-\beta _{1}+\cdots)q^{5}+\cdots$
422.4.a.c	$422$	$4$	422.a	1.a	$1$	$13$	$13$	$24.899$	$\mathbb{Q}[x]/(x^{13} - \cdots)$	None	✓	$1$	$1$	$-26$	$-5$	$-15$	$-56$	$-$	$2^{2}\cdot 5$	$\mathrm{SU}(2)$	$q-2q^{2}-\beta _{1}q^{3}+4q^{4}+(-1+\beta _{6}+\cdots)q^{5}+\cdots$
422.4.a.d	$422$	$4$	422.a	1.a	$1$	$14$	$14$	$24.899$	$\mathbb{Q}[x]/(x^{14} - \cdots)$	None	✓	$1$	$0$	$-28$	$4$	$20$	$56$	$+$	$2^{3}$	$\mathrm{SU}(2)$	$q-2q^{2}+\beta _{1}q^{3}+4q^{4}+(1+\beta _{6})q^{5}+\cdots$
422.4.a.e	$422$	$4$	422.a	1.a	$1$	$15$	$15$	$24.899$	$\mathbb{Q}[x]/(x^{15} - \cdots)$	None	✓	$1$	$0$	$30$	$24$	$15$	$86$	$+$	$2$	$\mathrm{SU}(2)$	$q+2q^{2}+(2-\beta _{1})q^{3}+4q^{4}+(1-\beta _{3}+\cdots)q^{5}+\cdots$
422.5.b.a	$422$	$5$	422.b	211.b	$2$	$72$	$72$	$43.622$		None		$2$	$0$	$0$	$0$	$0$	$0$			$\mathrm{SU}(2)[C_{2}]$
422.6.a.a	$422$	$6$	422.a	1.a	$1$	$20$	$20$	$67.682$	$\mathbb{Q}[x]/(x^{20} - \cdots)$	None	✓	$1$	$1$	$80$	$-59$	$-75$	$-565$	$+$	$2^{11}$	$\mathrm{SU}(2)$	$q+4q^{2}+(-3+\beta _{1})q^{3}+2^{4}q^{4}+(-4+\cdots)q^{5}+\cdots$
422.6.a.b	$422$	$6$	422.a	1.a	$1$	$21$	$21$	$67.682$		None	✓	$1$	$1$	$-84$	$-15$	$-100$	$-259$	$+$		$\mathrm{SU}(2)$
422.6.a.c	$422$	$6$	422.a	1.a	$1$	$22$	$22$	$67.682$		None	✓	$1$	$0$	$-88$	$12$	$75$	$525$	$-$		$\mathrm{SU}(2)$
422.6.a.d	$422$	$6$	422.a	1.a	$1$	$24$	$24$	$67.682$		None	✓	$1$	$0$	$96$	$58$	$100$	$611$	$-$		$\mathrm{SU}(2)$
422.8.a.a	$422$	$8$	422.a	1.a	$1$	$29$	$29$	$131.826$		None	✓	$1$	$1$	$232$	$-204$	$-500$	$-3861$	$-$		$\mathrm{SU}(2)$
422.8.a.b	$422$	$8$	422.a	1.a	$1$	$30$	$30$	$131.826$		None	✓	$1$	$1$	$-240$	$-38$	$-165$	$-4067$	$-$		$\mathrm{SU}(2)$
422.8.a.c	$422$	$8$	422.a	1.a	$1$	$31$	$31$	$131.826$		None	✓	$1$	$0$	$-248$	$43$	$710$	$1421$	$+$		$\mathrm{SU}(2)$
422.8.a.d	$422$	$8$	422.a	1.a	$1$	$33$	$33$	$131.826$		None	✓	$1$	$0$	$264$	$147$	$375$	$4371$	$+$		$\mathrm{SU}(2)$

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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.