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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 (48 matches)

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Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	Image	CM	RM	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	Image	CM	RM	Self-dual	Inner twists	Rank*	$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$	Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
1539.1.c.a	$1539$	$1$	1539.c	19.b	$2$	$1$	$1$	$0.768$	$\Q$	$D_{3}$	$\Q(\sqrt{-19}) $	None	✓	$2$	$0$	$0$	$0$	$-2$	$-1$		$1$		$q+q^{4}-2q^{5}-q^{7}+q^{11}+q^{16}+q^{17}+\cdots$
1539.1.c.b	$1539$	$1$	1539.c	19.b	$2$	$1$	$1$	$0.768$	$\Q$	$D_{3}$	$\Q(\sqrt{-19}) $	None	✓	$2$	$0$	$0$	$0$	$2$	$-1$		$1$		$q+q^{4}+2q^{5}-q^{7}-q^{11}+q^{16}-q^{17}+\cdots$
1539.1.c.c	$1539$	$1$	1539.c	19.b	$2$	$2$	$2$	$0.768$	$\Q(\sqrt{-1}) $	$A_{4}$	None	None		$2$	$0$	$0$	$0$	$-2$	$2$		$1$		$q-iq^{2}-q^{5}+q^{7}-iq^{8}+iq^{10}+\cdots$
1539.1.c.d	$1539$	$1$	1539.c	19.b	$2$	$2$	$2$	$0.768$	$\Q(\sqrt{-1}) $	$A_{4}$	None	None		$2$	$0$	$0$	$0$	$2$	$2$		$1$		$q-iq^{2}+q^{5}+q^{7}-iq^{8}-iq^{10}+\cdots$
1539.1.c.e	$1539$	$1$	1539.c	19.b	$2$	$2$	$2$	$0.768$	$\Q(\sqrt{3}) $	$D_{6}$	$\Q(\sqrt{-19}) $	None	✓	$4$	$0$	$0$	$0$	$0$	$2$		$1$		$q+q^{4}+q^{7}-\beta q^{11}+q^{16}+\beta q^{17}+\cdots$
1539.1.i.a	$1539$	$1$	1539.i	171.i	$6$	$2$	$1$	$0.768$	$\Q(\sqrt{-3}) $	$D_{6}$	$\Q(\sqrt{-3}) $	None		$4$	$0$	$0$	$0$	$0$	$-1$		$1$		$q+q^{4}-\zeta_{6}q^{7}+(-\zeta_{6}-\zeta_{6}^{2})q^{13}+\cdots$
1539.1.j.a	$1539$	$1$	1539.j	171.j	$6$	$2$	$1$	$0.768$	$\Q(\sqrt{-3}) $	$D_{3}$	$\Q(\sqrt{-3}) $	None		$4$	$0$	$0$	$0$	$0$	$1$		$1$		$q+q^{4}-\zeta_{6}^{2}q^{7}-q^{13}+q^{16}+q^{19}+\cdots$
1539.1.n.a	$1539$	$1$	1539.n	171.n	$6$	$2$	$1$	$0.768$	$\Q(\sqrt{-3}) $	$D_{3}$	$\Q(\sqrt{-3}) $	None		$4$	$0$	$0$	$0$	$0$	$1$		$1$		$q-\zeta_{6}q^{4}+\zeta_{6}q^{7}+\zeta_{6}q^{13}+\zeta_{6}^{2}q^{16}+\cdots$
1539.1.o.a	$1539$	$1$	1539.o	171.o	$6$	$2$	$1$	$0.768$	$\Q(\sqrt{-3}) $	$D_{3}$	$\Q(\sqrt{-19}) $	None		$4$	$0$	$0$	$0$	$-2$	$1$		$1$		$q+\zeta_{6}^{2}q^{4}+\zeta_{6}^{2}q^{5}+\zeta_{6}q^{7}+\zeta_{6}q^{11}+\cdots$
1539.1.o.b	$1539$	$1$	1539.o	171.o	$6$	$2$	$1$	$0.768$	$\Q(\sqrt{-3}) $	$D_{2}$	$\Q(\sqrt{-3}) $, $\Q(\sqrt{-19}) $	$\Q(\sqrt{57}) $		$8$	$0$	$0$	$0$	$0$	$2$		$1$		$q+\zeta_{6}^{2}q^{4}+\zeta_{6}q^{7}-\zeta_{6}q^{16}-q^{19}+\cdots$
1539.1.o.c	$1539$	$1$	1539.o	171.o	$6$	$2$	$1$	$0.768$	$\Q(\sqrt{-3}) $	$D_{3}$	$\Q(\sqrt{-19}) $	None		$4$	$0$	$0$	$0$	$2$	$1$		$1$		$q+\zeta_{6}^{2}q^{4}-\zeta_{6}^{2}q^{5}+\zeta_{6}q^{7}-\zeta_{6}q^{11}+\cdots$
1539.1.o.d	$1539$	$1$	1539.o	171.o	$6$	$4$	$2$	$0.768$	$\Q(\zeta_{12})$	$D_{6}$	$\Q(\sqrt{-19}) $	None		$8$	$0$	$0$	$0$	$0$	$-2$		$3$		$q-\zeta_{12}^{2}q^{4}+\zeta_{12}^{4}q^{7}+(-\zeta_{12}^{3}-\zeta_{12}^{5}+\cdots)q^{11}+\cdots$
1539.1.s.a	$1539$	$1$	1539.s	171.s	$6$	$2$	$1$	$0.768$	$\Q(\sqrt{-3}) $	$D_{6}$	$\Q(\sqrt{-3}) $	None		$4$	$0$	$0$	$0$	$0$	$-1$		$1$		$q+\zeta_{6}^{2}q^{4}+\zeta_{6}^{2}q^{7}+(1+\zeta_{6})q^{13}+\cdots$
1539.1.bq.a	$1539$	$1$	1539.bq	171.z	$18$	$6$	$1$	$0.768$	$\Q(\zeta_{18})$	$D_{9}$	$\Q(\sqrt{-3}) $	None		$4$	$0$	$0$	$0$	$0$	$0$		$1$		$q-\zeta_{18}q^{4}+(-\zeta_{18}+\zeta_{18}^{2})q^{7}+(\zeta_{18}^{6}+\cdots)q^{13}+\cdots$
1539.1.bt.a	$1539$	$1$	1539.bt	171.ac	$18$	$6$	$1$	$0.768$	$\Q(\zeta_{18})$	$D_{18}$	$\Q(\sqrt{-3}) $	None		$4$	$0$	$0$	$0$	$0$	$0$		$1$		$q-\zeta_{18}q^{4}+(\zeta_{18}-\zeta_{18}^{2})q^{7}+(-\zeta_{18}^{6}+\cdots)q^{13}+\cdots$
1539.1.ce.a	$1539$	$1$	1539.ce	171.ae	$18$	$6$	$1$	$0.768$	$\Q(\zeta_{18})$	$D_{18}$	$\Q(\sqrt{-3}) $	None		$4$	$0$	$0$	$0$	$0$	$0$		$1$		$q-\zeta_{18}^{7}q^{4}+(-\zeta_{18}^{4}+\zeta_{18}^{5})q^{7}+\cdots$
1539.1.cf.a	$1539$	$1$	1539.cf	171.af	$18$	$6$	$1$	$0.768$	$\Q(\zeta_{18})$	$D_{9}$	$\Q(\sqrt{-3}) $	None		$4$	$0$	$0$	$0$	$0$	$0$		$1$		$q-\zeta_{18}^{7}q^{4}+(\zeta_{18}^{4}-\zeta_{18}^{5})q^{7}+(-\zeta_{18}^{3}+\cdots)q^{13}+\cdots$
1539.2.a.a	$1539$	$2$	1539.a	1.a	$1$	$1$	$1$	$12.289$	$\Q$	$_{}$	None	None	✓	$1$	$1$	$-2$	$0$	$1$	$-3$	$+$	$1$	$\mathrm{SU}(2)$	$q-2q^{2}+2q^{4}+q^{5}-3q^{7}-2q^{10}+\cdots$
1539.2.a.b	$1539$	$2$	1539.a	1.a	$1$	$1$	$1$	$12.289$	$\Q$	$_{}$	None	None	✓	$1$	$1$	$-1$	$0$	$-3$	$-2$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-q^{4}-3q^{5}-2q^{7}+3q^{8}+3q^{10}+\cdots$
1539.2.a.c	$1539$	$2$	1539.a	1.a	$1$	$1$	$1$	$12.289$	$\Q$	$_{}$	None	None	✓	$1$	$1$	$-1$	$0$	$2$	$3$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-q^{4}+2q^{5}+3q^{7}+3q^{8}-2q^{10}+\cdots$
1539.2.a.d	$1539$	$2$	1539.a	1.a	$1$	$1$	$1$	$12.289$	$\Q$	$_{}$	None	None	✓	$1$	$1$	$1$	$0$	$-2$	$3$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-q^{4}-2q^{5}+3q^{7}-3q^{8}-2q^{10}+\cdots$
1539.2.a.e	$1539$	$2$	1539.a	1.a	$1$	$1$	$1$	$12.289$	$\Q$	$_{}$	None	None	✓	$1$	$1$	$1$	$0$	$3$	$-2$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-q^{4}+3q^{5}-2q^{7}-3q^{8}+3q^{10}+\cdots$
1539.2.a.f	$1539$	$2$	1539.a	1.a	$1$	$1$	$1$	$12.289$	$\Q$	$_{}$	None	None	✓	$1$	$1$	$2$	$0$	$-1$	$-3$	$+$	$1$	$\mathrm{SU}(2)$	$q+2q^{2}+2q^{4}-q^{5}-3q^{7}-2q^{10}+\cdots$
1539.2.a.g	$1539$	$2$	1539.a	1.a	$1$	$2$	$2$	$12.289$	$\Q(\sqrt{3}) $	$_{}$	None	None	✓	$1$	$0$	$-2$	$0$	$0$	$2$	$-$	$1$	$\mathrm{SU}(2)$	$q+(-1+\beta )q^{2}+(2-2\beta )q^{4}+\beta q^{5}+\cdots$
1539.2.a.h	$1539$	$2$	1539.a	1.a	$1$	$2$	$2$	$12.289$	$\Q(\sqrt{3}) $	$_{}$	None	None	✓	$2$	$1$	$0$	$0$	$0$	$-2$	$+$	$1$	$\mathrm{SU}(2)$	$q-2q^{4}+\beta q^{5}-q^{7}-2\beta q^{11}+2q^{13}+\cdots$
1539.2.a.i	$1539$	$2$	1539.a	1.a	$1$	$2$	$2$	$12.289$	$\Q(\sqrt{3}) $	$_{}$	None	None	✓	$2$	$1$	$0$	$0$	$0$	$-2$	$+$	$1$	$\mathrm{SU}(2)$	$q+\beta q^{2}+q^{4}-q^{7}-\beta q^{8}-\beta q^{11}+\cdots$
1539.2.a.j	$1539$	$2$	1539.a	1.a	$1$	$2$	$2$	$12.289$	$\Q(\sqrt{3}) $	$_{}$	None	None	✓	$1$	$0$	$2$	$0$	$0$	$2$	$-$	$1$	$\mathrm{SU}(2)$	$q+(1+\beta )q^{2}+(2+2\beta )q^{4}+\beta q^{5}+(1+\cdots)q^{7}+\cdots$
1539.2.a.k	$1539$	$2$	1539.a	1.a	$1$	$7$	$7$	$12.289$	$\mathbb{Q}[x]/(x^{7} - \cdots)$	$_{}$	None	None	✓	$1$	$0$	$0$	$0$	$0$	$5$	$-$	$1$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}-\beta _{4}q^{5}+(1+\beta _{1}+\cdots)q^{7}+\cdots$
1539.2.a.l	$1539$	$2$	1539.a	1.a	$1$	$7$	$7$	$12.289$	$\mathbb{Q}[x]/(x^{7} - \cdots)$	$_{}$	None	None	✓	$1$	$0$	$0$	$0$	$0$	$5$	$-$	$1$	$\mathrm{SU}(2)$	$q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}+\beta _{4}q^{5}+(1+\beta _{1}+\cdots)q^{7}+\cdots$
1539.2.a.m	$1539$	$2$	1539.a	1.a	$1$	$8$	$8$	$12.289$	8.8.$\cdots$.1	$_{}$	None	None	✓	$2$	$0$	$0$	$0$	$0$	$-6$	$-$	$2$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(\beta _{5}+\beta _{7})q^{5}+\cdots$
1539.2.a.n	$1539$	$2$	1539.a	1.a	$1$	$9$	$9$	$12.289$	$\mathbb{Q}[x]/(x^{9} - \cdots)$	$_{}$	None	None	✓	$1$	$1$	$-3$	$0$	$-9$	$0$	$+$	$3$	$\mathrm{SU}(2)$	$q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(-1-\beta _{8})q^{5}+\cdots$
1539.2.a.o	$1539$	$2$	1539.a	1.a	$1$	$9$	$9$	$12.289$	$\mathbb{Q}[x]/(x^{9} - \cdots)$	$_{}$	None	None	✓	$1$	$1$	$-3$	$0$	$-7$	$0$	$+$	$1$	$\mathrm{SU}(2)$	$q-\beta _{1}q^{2}+(1+\beta _{1}+\beta _{2})q^{4}+(-1+\beta _{7}+\cdots)q^{5}+\cdots$
1539.2.a.p	$1539$	$2$	1539.a	1.a	$1$	$9$	$9$	$12.289$	$\mathbb{Q}[x]/(x^{9} - \cdots)$	$_{}$	None	None	✓	$1$	$0$	$3$	$0$	$7$	$0$	$-$	$1$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{2}+(1+\beta _{1}+\beta _{2})q^{4}+(1-\beta _{7}+\cdots)q^{5}+\cdots$
1539.2.a.q	$1539$	$2$	1539.a	1.a	$1$	$9$	$9$	$12.289$	$\mathbb{Q}[x]/(x^{9} - \cdots)$	$_{}$	None	None	✓	$1$	$0$	$3$	$0$	$9$	$0$	$-$	$3$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(1+\beta _{8})q^{5}+\cdots$
1539.2.d.a	$1539$	$2$	1539.d	57.d	$2$	$8$	$8$	$12.289$	8.0.$\cdots$.4	$_{}$	$\Q(\sqrt{-19}) $	None		$4$	$0$	$0$	$0$	$0$	$0$		$3^{5}$	$\mathrm{U}(1)[D_{2}]$	$q-2q^{4}-\beta _{5}q^{5}+\beta _{1}q^{7}-\beta _{7}q^{11}+\cdots$
1539.2.d.b	$1539$	$2$	1539.d	57.d	$2$	$16$	$16$	$12.289$	$\mathbb{Q}[x]/(x^{16} - \cdots)$	$_{}$	None	None		$4$	$0$	$0$	$0$	$0$	$-8$		$2^{8}$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta _{3}q^{2}+(2+\beta _{8}-\beta _{11})q^{4}+(-\beta _{6}+\cdots)q^{5}+\cdots$
1539.2.d.c	$1539$	$2$	1539.d	57.d	$2$	$16$	$16$	$12.289$	$\mathbb{Q}[x]/(x^{16} - \cdots)$	$_{}$	None	None		$4$	$0$	$0$	$0$	$0$	$0$		$2^{7}$	$\mathrm{SU}(2)[C_{2}]$	$q-\beta _{8}q^{2}+(2+\beta _{3}-\beta _{13})q^{4}+\beta _{5}q^{5}+\cdots$
1539.2.d.d	$1539$	$2$	1539.d	57.d	$2$	$36$	$36$	$12.289$		$_{}$	None	None		$4$	$0$	$0$	$0$	$0$	$8$			$\mathrm{SU}(2)[C_{2}]$
1539.4.a.a	$1539$	$4$	1539.a	1.a	$1$	$11$	$11$	$90.804$	$\mathbb{Q}[x]/(x^{11} - \cdots)$	$_{}$	None	None	✓	$1$	$1$	$-3$	$0$	$-6$	$-11$	$-$	$2^{4}\cdot 3^{3}$	$\mathrm{SU}(2)$	$q-\beta _{1}q^{2}+(2+\beta _{2})q^{4}+(-1+\beta _{5})q^{5}+\cdots$
1539.4.a.b	$1539$	$4$	1539.a	1.a	$1$	$11$	$11$	$90.804$	$\mathbb{Q}[x]/(x^{11} - \cdots)$	$_{}$	None	None	✓	$1$	$1$	$3$	$0$	$6$	$-11$	$-$	$2^{4}\cdot 3^{3}$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{2}+(2+\beta _{2})q^{4}+(1-\beta _{5})q^{5}+\cdots$
1539.4.a.c	$1539$	$4$	1539.a	1.a	$1$	$15$	$15$	$90.804$	$\mathbb{Q}[x]/(x^{15} - \cdots)$	$_{}$	None	None	✓	$1$	$0$	$-3$	$0$	$22$	$-19$	$+$	$2^{5}\cdot 3^{4}$	$\mathrm{SU}(2)$	$q-\beta _{1}q^{2}+(4+\beta _{1}+\beta _{2})q^{4}+(1-\beta _{7}+\cdots)q^{5}+\cdots$
1539.4.a.d	$1539$	$4$	1539.a	1.a	$1$	$15$	$15$	$90.804$	$\mathbb{Q}[x]/(x^{15} - \cdots)$	$_{}$	None	None	✓	$1$	$0$	$3$	$0$	$-22$	$-19$	$+$	$2^{5}\cdot 3^{4}$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{2}+(4+\beta _{1}+\beta _{2})q^{4}+(-1+\beta _{7}+\cdots)q^{5}+\cdots$
1539.4.a.e	$1539$	$4$	1539.a	1.a	$1$	$24$	$24$	$90.804$		$_{}$	None	None	✓	$2$	$1$	$0$	$0$	$0$	$-34$	$-$		$\mathrm{SU}(2)$
1539.4.a.f	$1539$	$4$	1539.a	1.a	$1$	$27$	$27$	$90.804$		$_{}$	None	None	✓	$1$	$1$	$-6$	$0$	$-54$	$0$	$-$		$\mathrm{SU}(2)$
1539.4.a.g	$1539$	$4$	1539.a	1.a	$1$	$27$	$27$	$90.804$		$_{}$	None	None	✓	$1$	$1$	$-6$	$0$	$-26$	$0$	$-$		$\mathrm{SU}(2)$
1539.4.a.h	$1539$	$4$	1539.a	1.a	$1$	$27$	$27$	$90.804$		$_{}$	None	None	✓	$1$	$0$	$6$	$0$	$26$	$0$	$+$		$\mathrm{SU}(2)$
1539.4.a.i	$1539$	$4$	1539.a	1.a	$1$	$27$	$27$	$90.804$		$_{}$	None	None	✓	$1$	$0$	$6$	$0$	$54$	$0$	$+$		$\mathrm{SU}(2)$
1539.4.a.j	$1539$	$4$	1539.a	1.a	$1$	$32$	$32$	$90.804$		$_{}$	None	None	✓	$2$	$0$	$0$	$0$	$0$	$94$	$+$		$\mathrm{SU}(2)$

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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
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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.