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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 (1-50 of 81 matches)

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Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	Image	CM	RM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	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	Image	CM	RM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$	Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
209.1.h.a	$209$	$1$	209.h	209.h	$6$	$4$	$2$	$0.104$	$\Q(\zeta_{12})$	$A_{4}$	None	None		✓	✓	✓	209.1.h.a	$4$	$0$	$0$	$-2$	$-2$	$0$		$1$		$q+\zeta_{12}^{5}q^{2}-\zeta_{12}^{2}q^{3}-\zeta_{12}^{2}q^{5}+\cdots$
209.1.m.a	$209$	$1$	209.m	209.m	$10$	$4$	$1$	$0.104$	$\Q(\zeta_{10})$	$D_{5}$	$\Q(\sqrt{-19}) $	None		✓	✓	✓	209.1.m.a	$4$	$0$	$0$	$0$	$-2$	$3$		$1$		$q+\zeta_{10}^{4}q^{4}+(\zeta_{10}^{2}+\zeta_{10}^{4})q^{5}+(1+\cdots)q^{7}+\cdots$
209.2.a.a	$209$	$2$	209.a	1.a	$1$	$1$	$1$	$1.669$	$\Q$	$_{}$	None	None	✓	✓	✓	✓	209.2.a.a	$1$	$1$	$0$	$1$	$-3$	$-4$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{3}-2q^{4}-3q^{5}-4q^{7}-2q^{9}+\cdots$
209.2.a.b	$209$	$2$	209.a	1.a	$1$	$2$	$2$	$1.669$	$\Q(\sqrt{2}) $	$_{}$	None	None	✓	✓	✓	✓	209.2.a.b	$1$	$1$	$0$	$-2$	$-2$	$-4$	$+$	$1$	$\mathrm{SU}(2)$	$q+\beta q^{2}+(-1-\beta )q^{3}-q^{5}+(-2-\beta )q^{6}+\cdots$
209.2.a.c	$209$	$2$	209.a	1.a	$1$	$5$	$5$	$1.669$	5.5.246832.1	$_{}$	None	None	✓	✓	✓	✓	209.2.a.c	$1$	$0$	$2$	$1$	$-5$	$6$	$-$	$1$	$\mathrm{SU}(2)$	$q+\beta _{2}q^{2}+\beta _{3}q^{3}+(\beta _{1}+\beta _{2}+\beta _{3}+\beta _{4})q^{4}+\cdots$
209.2.a.d	$209$	$2$	209.a	1.a	$1$	$7$	$7$	$1.669$	$\mathbb{Q}[x]/(x^{7} - \cdots)$	$_{}$	None	None	✓	✓	✓	✓	209.2.a.d	$1$	$0$	$-1$	$2$	$2$	$10$	$-$	$1$	$\mathrm{SU}(2)$	$q-\beta _{1}q^{2}-\beta _{2}q^{3}+(2+\beta _{2}+\beta _{3})q^{4}+\cdots$
209.2.d.a	$209$	$2$	209.d	209.d	$2$	$2$	$2$	$1.669$	$\Q(\sqrt{-19}) $	$_{}$	$\Q(\sqrt{-19}) $	None		✓			209.2.d.a	$4$	$0$	$0$	$0$	$2$	$0$		$1$	$\mathrm{U}(1)[D_{2}]$	$q-2q^{4}+q^{5}+(1-2\beta )q^{7}+3q^{9}+(3+\cdots)q^{11}+\cdots$
209.2.d.b	$209$	$2$	209.d	209.d	$2$	$8$	$8$	$1.669$	8.0.3288334336.3	$_{}$	None	None		✓			209.2.d.b	$4$	$0$	$0$	$0$	$-8$	$0$		$2$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta _{7}q^{2}+\beta _{2}q^{3}-\beta _{3}q^{4}+(-1-\beta _{3}+\cdots)q^{5}+\cdots$
209.2.d.c	$209$	$2$	209.d	209.d	$2$	$8$	$8$	$1.669$	8.0.484000000.9	$_{}$	None	None		✓			209.2.d.c	$4$	$0$	$0$	$0$	$4$	$0$		$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta _{5}q^{2}+\beta _{2}q^{3}+(1+2\beta _{3})q^{4}+(2+\cdots)q^{5}+\cdots$
209.2.e.a	$209$	$2$	209.e	19.c	$3$	$18$	$9$	$1.669$	$\mathbb{Q}[x]/(x^{18} - \cdots)$	$_{}$	None	None		✓			209.2.e.a	$2$	$0$	$-1$	$-4$	$0$	$-6$		$3^{2}$	$\mathrm{SU}(2)[C_{3}]$	$q-\beta _{1}q^{2}-\beta _{7}q^{3}+(-1-\beta _{4}-\beta _{9}+\cdots)q^{4}+\cdots$
209.2.e.b	$209$	$2$	209.e	19.c	$3$	$18$	$9$	$1.669$	$\mathbb{Q}[x]/(x^{18} - \cdots)$	$_{}$	None	None		✓			209.2.e.b	$2$	$0$	$1$	$0$	$0$	$-6$		$3$	$\mathrm{SU}(2)[C_{3}]$	$q+\beta _{1}q^{2}+(\beta _{4}-\beta _{12})q^{3}+(-1+\beta _{3}+\cdots)q^{4}+\cdots$
209.2.f.a	$209$	$2$	209.f	11.c	$5$	$4$	$1$	$1.669$	$\Q(\zeta_{10})$	$_{}$	None	None		✓			209.2.f.a	$2$	$1$	$-5$	$-6$	$-2$	$-1$		$1$	$\mathrm{SU}(2)[C_{5}]$	$q+(-1-\zeta_{10}+\zeta_{10}^{2}+\zeta_{10}^{3})q^{2}+\cdots$
209.2.f.b	$209$	$2$	209.f	11.c	$5$	$28$	$7$	$1.669$		$_{}$	None	None		✓			209.2.f.b	$2$	$0$	$-2$	$1$	$7$	$-12$			$\mathrm{SU}(2)[C_{5}]$
209.2.f.c	$209$	$2$	209.f	11.c	$5$	$40$	$10$	$1.669$		$_{}$	None	None		✓	✓		209.2.f.c	$2$	$0$	$3$	$3$	$-9$	$4$			$\mathrm{SU}(2)[C_{5}]$
209.2.g.a	$209$	$2$	209.g	209.g	$6$	$16$	$8$	$1.669$	16.0.$\cdots$.1	$_{}$	None	None		✓			209.2.g.a	$4$	$0$	$0$	$-12$	$8$	$0$		$2^{12}$	$\mathrm{SU}(2)[C_{6}]$	$q+(-\beta _{1}+\beta _{5})q^{2}+(-1+\beta _{6})q^{3}+(-1+\cdots)q^{4}+\cdots$
209.2.g.b	$209$	$2$	209.g	209.g	$6$	$20$	$10$	$1.669$	$\mathbb{Q}[x]/(x^{20} + \cdots)$	$_{}$	None	None		✓	✓		209.2.g.b	$4$	$0$	$0$	$0$	$-12$	$0$		$1$	$\mathrm{SU}(2)[C_{6}]$	$q+(\beta _{1}-\beta _{6})q^{2}-\beta _{12}q^{3}+(-1+\beta _{3}+\cdots)q^{4}+\cdots$
209.2.j.a	$209$	$2$	209.j	19.e	$9$	$6$	$1$	$1.669$	$\Q(\zeta_{18})$	$_{}$	None	None		✓			209.2.j.a	$2$	$0$	$3$	$0$	$6$	$6$		$1$	$\mathrm{SU}(2)[C_{9}]$	$q+(1-\zeta_{18}+\zeta_{18}^{2}-\zeta_{18}^{3}+\zeta_{18}^{4}+\cdots)q^{2}+\cdots$
209.2.j.b	$209$	$2$	209.j	19.e	$9$	$42$	$7$	$1.669$		$_{}$	None	None		✓			209.2.j.b	$2$	$0$	$-3$	$0$	$-6$	$-6$			$\mathrm{SU}(2)[C_{9}]$
209.2.j.c	$209$	$2$	209.j	19.e	$9$	$48$	$8$	$1.669$		$_{}$	None	None		✓	✓		209.2.j.c	$2$	$0$	$0$	$0$	$0$	$0$			$\mathrm{SU}(2)[C_{9}]$
209.2.k.a	$209$	$2$	209.k	209.k	$10$	$8$	$2$	$1.669$	8.0.2036265625.1	$_{}$	$\Q(\sqrt{-19}) $	None		✓			209.2.k.a	$4$	$0$	$0$	$0$	$-2$	$-15$		$1$	$\mathrm{U}(1)[D_{10}]$	$q-2\beta _{6}q^{4}+(\beta _{1}-\beta _{3}+\beta _{6})q^{5}+(-1+\cdots)q^{7}+\cdots$
209.2.k.b	$209$	$2$	209.k	209.k	$10$	$8$	$2$	$1.669$	8.0.484000000.9	$_{}$	None	None		✓			209.2.k.b	$4$	$0$	$0$	$0$	$4$	$-10$		$1$	$\mathrm{SU}(2)[C_{10}]$	$q+(-\beta _{1}-\beta _{4})q^{2}+(-1-\beta _{2}-3\beta _{3}+\cdots)q^{4}+\cdots$
209.2.k.c	$209$	$2$	209.k	209.k	$10$	$56$	$14$	$1.669$		$_{}$	None	None		✓	✓		209.2.k.c	$4$	$0$	$0$	$0$	$-10$	$20$			$\mathrm{SU}(2)[C_{10}]$
209.2.n.a	$209$	$2$	209.n	209.n	$15$	$144$	$18$	$1.669$		$_{}$	None	None		✓	✓	✓	209.2.n.a	$4$	$0$	$-5$	$-1$	$-5$	$-18$			$\mathrm{SU}(2)[C_{15}]$
209.2.p.a	$209$	$2$	209.p	209.p	$18$	$108$	$18$	$1.669$		$_{}$	None	None		✓	✓	✓	209.2.p.a	$4$	$0$	$0$	$-6$	$-12$	$0$			$\mathrm{SU}(2)[C_{18}]$
209.2.t.a	$209$	$2$	209.t	209.t	$30$	$144$	$18$	$1.669$		$_{}$	None	None		✓	✓	✓	209.2.t.a	$4$	$0$	$-15$	$-3$	$-1$	$-10$			$\mathrm{SU}(2)[C_{30}]$
209.2.u.a	$209$	$2$	209.u	209.u	$45$	$432$	$18$	$1.669$		$_{}$	None	None		✓	✓	✓	209.2.u.a	$4$	$0$	$-18$	$-24$	$-18$	$0$			$\mathrm{SU}(2)[C_{45}]$
209.2.w.a	$209$	$2$	209.w	209.w	$90$	$432$	$18$	$1.669$		$_{}$	None	None		✓	✓	✓	209.2.w.a	$4$	$0$	$-30$	$-24$	$-18$	$-30$			$\mathrm{SU}(2)[C_{90}]$
209.3.b.a	$209$	$3$	209.b	19.b	$2$	$32$	$32$	$5.695$		$_{}$	None	None		✓	✓	✓	209.3.b.a	$2$	$0$	$0$	$0$	$0$	$20$			$\mathrm{SU}(2)[C_{2}]$
209.3.c.a	$209$	$3$	209.c	11.b	$2$	$36$	$36$	$5.695$		$_{}$	None	None		✓	✓	✓	209.3.c.a	$2$	$0$	$0$	$-2$	$-4$	$0$			$\mathrm{SU}(2)[C_{2}]$
209.3.h.a	$209$	$3$	209.h	209.h	$6$	$76$	$38$	$5.695$		$_{}$	None	None		✓	✓	✓	209.3.h.a	$4$	$0$	$0$	$-6$	$0$	$0$			$\mathrm{SU}(2)[C_{6}]$
209.3.i.a	$209$	$3$	209.i	19.d	$6$	$64$	$32$	$5.695$		$_{}$	None	None		✓	✓	✓	209.3.i.a	$2$	$0$	$0$	$0$	$0$	$-20$			$\mathrm{SU}(2)[C_{6}]$
209.3.l.a	$209$	$3$	209.l	11.d	$10$	$144$	$36$	$5.695$		$_{}$	None	None		✓	✓	✓	209.3.l.a	$2$	$0$	$0$	$2$	$4$	$-15$			$\mathrm{SU}(2)[C_{10}]$
209.3.m.a	$209$	$3$	209.m	209.m	$10$	$8$	$2$	$5.695$	8.0.2036265625.1	$_{}$	$\Q(\sqrt{-19}) $	None		✓			209.3.m.a	$4$	$0$	$0$	$0$	$18$	$-15$		$5$	$\mathrm{U}(1)[D_{10}]$	$q+4\beta _{1}q^{4}+(5+5\beta _{2}-4\beta _{3}+\beta _{5}-\beta _{6}+\cdots)q^{5}+\cdots$
209.3.m.b	$209$	$3$	209.m	209.m	$10$	$144$	$36$	$5.695$		$_{}$	None	None		✓	✓		209.3.m.b	$4$	$0$	$0$	$0$	$-26$	$0$			$\mathrm{SU}(2)[C_{10}]$
209.3.o.a	$209$	$3$	209.o	19.f	$18$	$204$	$34$	$5.695$		$_{}$	None	None		✓	✓	✓	209.3.o.a	$2$	$0$	$0$	$0$	$0$	$0$			$\mathrm{SU}(2)[C_{18}]$
209.3.q.a	$209$	$3$	209.q	209.q	$18$	$228$	$38$	$5.695$		$_{}$	None	None		✓	✓	✓	209.3.q.a	$4$	$0$	$0$	$0$	$-12$	$0$			$\mathrm{SU}(2)[C_{18}]$
209.3.r.a	$209$	$3$	209.r	209.r	$30$	$304$	$38$	$5.695$		$_{}$	None	None		✓	✓	✓	209.3.r.a	$4$	$0$	$-9$	$3$	$-1$	$-30$			$\mathrm{SU}(2)[C_{30}]$
209.3.s.a	$209$	$3$	209.s	209.s	$30$	$304$	$38$	$5.695$		$_{}$	None	None		✓	✓	✓	209.3.s.a	$4$	$0$	$-5$	$1$	$-5$	$10$			$\mathrm{SU}(2)[C_{30}]$
209.3.v.a	$209$	$3$	209.v	209.v	$90$	$912$	$38$	$5.695$		$_{}$	None	None		✓	✓	✓	209.3.v.a	$4$	$0$	$-30$	$-30$	$-18$	$-60$			$\mathrm{SU}(2)[C_{90}]$
209.3.x.a	$209$	$3$	209.x	209.x	$90$	$912$	$38$	$5.695$		$_{}$	None	None		✓	✓	✓	209.3.x.a	$4$	$0$	$-18$	$-30$	$-18$	$18$			$\mathrm{SU}(2)[C_{90}]$
209.4.a.a	$209$	$4$	209.a	1.a	$1$	$8$	$8$	$12.331$	$\mathbb{Q}[x]/(x^{8} - \cdots)$	$_{}$	None	None	✓	✓	✓	✓	209.4.a.a	$1$	$1$	$0$	$-1$	$-12$	$-59$	$-$	$1$	$\mathrm{SU}(2)$	$q-\beta _{1}q^{2}+\beta _{3}q^{3}+(3+\beta _{2})q^{4}+(-2+\cdots)q^{5}+\cdots$
209.4.a.b	$209$	$4$	209.a	1.a	$1$	$10$	$10$	$12.331$	$\mathbb{Q}[x]/(x^{10} - \cdots)$	$_{}$	None	None	✓	✓	✓	✓	209.4.a.b	$1$	$1$	$-6$	$-9$	$-10$	$-53$	$-$	$2$	$\mathrm{SU}(2)$	$q+(-1+\beta _{1})q^{2}+(-1+\beta _{4})q^{3}+(2+\cdots)q^{4}+\cdots$
209.4.a.c	$209$	$4$	209.a	1.a	$1$	$13$	$13$	$12.331$	$\mathbb{Q}[x]/(x^{13} - \cdots)$	$_{}$	None	None	✓	✓	✓	✓	209.4.a.c	$1$	$0$	$-2$	$11$	$8$	$39$	$+$	$2^{2}$	$\mathrm{SU}(2)$	$q-\beta _{1}q^{2}+(1+\beta _{5})q^{3}+(6+\beta _{2})q^{4}+\cdots$
209.4.a.d	$209$	$4$	209.a	1.a	$1$	$15$	$15$	$12.331$	$\mathbb{Q}[x]/(x^{15} - \cdots)$	$_{}$	None	None	✓	✓	✓	✓	209.4.a.d	$1$	$0$	$4$	$3$	$10$	$73$	$+$	$2^{3}$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{2}+\beta _{3}q^{3}+(5+\beta _{2})q^{4}+(1+\beta _{6}+\cdots)q^{5}+\cdots$
209.4.d.a	$209$	$4$	209.d	209.d	$2$	$2$	$2$	$12.331$	$\Q(\sqrt{-19}) $	$_{}$	$\Q(\sqrt{-19}) $	None		✓			209.4.d.a	$4$	$0$	$0$	$0$	$-28$	$0$		$2$	$\mathrm{U}(1)[D_{2}]$	$q-8q^{4}-14q^{5}-2\beta q^{7}+3^{3}q^{9}+(-20+\cdots)q^{11}+\cdots$
209.4.d.b	$209$	$4$	209.d	209.d	$2$	$56$	$56$	$12.331$		$_{}$	None	None		✓	✓		209.4.d.b	$4$	$0$	$0$	$0$	$16$	$0$			$\mathrm{SU}(2)[C_{2}]$
209.4.e.a	$209$	$4$	209.e	19.c	$3$	$50$	$25$	$12.331$		$_{}$	None	None		✓			209.4.e.a	$2$	$0$	$-2$	$-6$	$0$	$-20$			$\mathrm{SU}(2)[C_{3}]$
209.4.e.b	$209$	$4$	209.e	19.c	$3$	$50$	$25$	$12.331$		$_{}$	None	None		✓			209.4.e.b	$2$	$0$	$2$	$6$	$0$	$-20$			$\mathrm{SU}(2)[C_{3}]$
209.4.f.a	$209$	$4$	209.f	11.c	$5$	$100$	$25$	$12.331$		$_{}$	None	None		✓			209.4.f.a	$2$	$0$	$2$	$8$	$26$	$119$			$\mathrm{SU}(2)[C_{5}]$
209.4.f.b	$209$	$4$	209.f	11.c	$5$	$116$	$29$	$12.331$		$_{}$	None	None		✓	✓		209.4.f.b	$2$	$0$	$2$	$-4$	$-26$	$-61$			$\mathrm{SU}(2)[C_{5}]$

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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.