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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 (50 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
729.1.f.a	$729$	$1$	729.f	27.f	$18$	$6$	$1$	$0.364$	$\Q(\zeta_{18})$	$D_{3}$	$\Q(\sqrt{-3})$	None			✓	✓	243.1.b.a	$12$	$0$	$0$	$0$	$0$	$0$		$1$		$q-\zeta_{18}q^{4}-\zeta_{18}^{8}q^{7}-\zeta_{18}^{4}q^{13}+\cdots$
729.2.a.a	$729$	$2$	729.a	1.a	$1$	$6$	$6$	$5.821$	6.6.1397493.1	$_{}$	None	None	✓				27.2.e.a	$1$	$1$	$-3$	$0$	$-6$	$0$	$+$	$1$	$\mathrm{SU}(2)$	$q+(-\beta _{1}-\beta _{5})q^{2}+(\beta _{1}-\beta _{2}-\beta _{3})q^{4}+\cdots$
729.2.a.b	$729$	$2$	729.a	1.a	$1$	$6$	$6$	$5.821$	6.6.7459857.1	$_{}$	None	None	✓	✓			729.2.a.b	$1$	$0$	$-3$	$0$	$3$	$6$	$-$	$1$	$\mathrm{SU}(2)$	$q+(-\beta _{1}-\beta _{3})q^{2}+(1+\beta _{1}+\beta _{2}+\beta _{4}+\cdots)q^{4}+\cdots$
729.2.a.c	$729$	$2$	729.a	1.a	$1$	$6$	$6$	$5.821$	$\Q(\zeta_{36})^+$	$_{}$	None	None	✓	✓			729.2.a.c	$2$	$1$	$0$	$0$	$0$	$-12$	$+$	$1$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{2}+\beta _{2}q^{4}+(-\beta _{1}-\beta _{3})q^{5}+\cdots$
729.2.a.d	$729$	$2$	729.a	1.a	$1$	$6$	$6$	$5.821$	6.6.1397493.1	$_{}$	None	None	✓				27.2.e.a	$1$	$0$	$3$	$0$	$6$	$0$	$-$	$1$	$\mathrm{SU}(2)$	$q+(\beta _{1}+\beta _{5})q^{2}+(\beta _{1}-\beta _{2}-\beta _{3})q^{4}+\cdots$
729.2.a.e	$729$	$2$	729.a	1.a	$1$	$6$	$6$	$5.821$	6.6.7459857.1	$_{}$	None	None	✓	✓			729.2.a.b	$1$	$0$	$3$	$0$	$-3$	$6$	$-$	$1$	$\mathrm{SU}(2)$	$q+(\beta _{1}+\beta _{3})q^{2}+(1+\beta _{1}+\beta _{2}+\beta _{4}+\cdots)q^{4}+\cdots$
729.2.c.a	$729$	$2$	729.c	9.c	$3$	$12$	$6$	$5.821$	$\mathbb{Q}[x]/(x^{12} + \cdots)$	$_{}$	None	None		✓			729.2.a.b	$2$	$0$	$-3$	$0$	$3$	$-6$		$3^{3}$	$\mathrm{SU}(2)[C_{3}]$	$q+(-\beta _{1}+\beta _{2}-\beta _{3}-\beta _{6})q^{2}+(-1+\cdots)q^{4}+\cdots$
729.2.c.b	$729$	$2$	729.c	9.c	$3$	$12$	$6$	$5.821$	12.0. $\cdots$ .1	$_{}$	None	None					27.2.e.a	$2$	$0$	$-3$	$0$	$-6$	$0$		$3^{3}$	$\mathrm{SU}(2)[C_{3}]$	$q+(\beta _{6}+\beta _{8}-\beta _{10})q^{2}+(-1+\beta _{2}-\beta _{3}+\cdots)q^{4}+\cdots$
729.2.c.c	$729$	$2$	729.c	9.c	$3$	$12$	$6$	$5.821$	$\Q(\zeta_{36})$	$_{}$	None	None		✓			729.2.a.c	$4$	$0$	$0$	$0$	$0$	$12$		$3^{3}$	$\mathrm{SU}(2)[C_{3}]$	$q+(\beta_{10}-\beta_{8}-\beta_{7})q^{2}-\beta_{11} q^{4}+(\beta_{10}+\beta_{5}-\beta_{2})q^{5}+\cdots$
729.2.c.d	$729$	$2$	729.c	9.c	$3$	$12$	$6$	$5.821$	$\mathbb{Q}[x]/(x^{12} + \cdots)$	$_{}$	None	None		✓			729.2.a.b	$2$	$0$	$3$	$0$	$-3$	$-6$		$3^{3}$	$\mathrm{SU}(2)[C_{3}]$	$q+(\beta _{2}-\beta _{3}+\beta _{5}-\beta _{6}+\beta _{9})q^{2}+(-1+\cdots)q^{4}+\cdots$
729.2.c.e	$729$	$2$	729.c	9.c	$3$	$12$	$6$	$5.821$	12.0. $\cdots$ .1	$_{}$	None	None					27.2.e.a	$2$	$0$	$3$	$0$	$6$	$0$		$3^{3}$	$\mathrm{SU}(2)[C_{3}]$	$q+(-\beta _{6}-\beta _{8}+\beta _{10})q^{2}+(-1+\beta _{2}+\cdots)q^{4}+\cdots$
729.2.e.a	$729$	$2$	729.e	27.e	$9$	$6$	$1$	$5.821$	$\Q(\zeta_{18})$	$_{}$	None	None					243.2.a.e	$2$	$1$	$-6$	$0$	$-3$	$-9$		$1$	$\mathrm{SU}(2)[C_{9}]$	$q+(-1-\zeta_{18}+\zeta_{18}^{4}+\zeta_{18}^{5})q^{2}+\cdots$
729.2.e.b	$729$	$2$	729.e	27.e	$9$	$6$	$1$	$5.821$	$\Q(\zeta_{18})$	$_{}$	None	None					243.2.a.e	$2$	$0$	$-3$	$0$	$-6$	$9$		$1$	$\mathrm{SU}(2)[C_{9}]$	$q+(-1+\zeta_{18}-\zeta_{18}^{2}+\zeta_{18}^{3}-\zeta_{18}^{4}+\cdots)q^{2}+\cdots$
729.2.e.c	$729$	$2$	729.e	27.e	$9$	$6$	$1$	$5.821$	$\Q(\zeta_{18})$	$_{}$	None	None					243.2.a.e	$2$	$0$	$-3$	$0$	$3$	$0$		$1$	$\mathrm{SU}(2)[C_{9}]$	$q+(\zeta_{18}+\zeta_{18}^{2}-\zeta_{18}^{3}-\zeta_{18}^{4})q^{2}+\cdots$
729.2.e.d	$729$	$2$	729.e	27.e	$9$	$6$	$1$	$5.821$	$\Q(\zeta_{18})$	$_{}$	$\Q(\sqrt{-3})$	None					243.2.a.b	$12$	$0$	$0$	$0$	$0$	$0$		$1$	$\mathrm{U}(1)[D_{9}]$	$q+2\zeta_{18}^{5}q^{4}+5\zeta_{18}^{4}q^{7}+2\zeta_{18}^{2}q^{13}+\cdots$
729.2.e.e	$729$	$2$	729.e	27.e	$9$	$6$	$1$	$5.821$	$\Q(\zeta_{18})$	$_{}$	$\Q(\sqrt{-3})$	None					243.2.a.a	$12$	$0$	$0$	$0$	$0$	$0$		$1$	$\mathrm{U}(1)[D_{9}]$	$q+2\zeta_{18}^{5}q^{4}-4\zeta_{18}^{4}q^{7}-7\zeta_{18}^{2}q^{13}+\cdots$
729.2.e.f	$729$	$2$	729.e	27.e	$9$	$6$	$1$	$5.821$	$\Q(\zeta_{18})$	$_{}$	$\Q(\sqrt{-3})$	None					27.2.a.a	$12$	$0$	$0$	$0$	$0$	$0$		$1$	$\mathrm{U}(1)[D_{9}]$	$q+2\zeta_{18}^{5}q^{4}-\zeta_{18}^{4}q^{7}+5\zeta_{18}^{2}q^{13}+\cdots$
729.2.e.g	$729$	$2$	729.e	27.e	$9$	$6$	$1$	$5.821$	$\Q(\zeta_{18})$	$_{}$	None	None					243.2.a.e	$2$	$0$	$3$	$0$	$6$	$9$		$1$	$\mathrm{SU}(2)[C_{9}]$	$q+(1-\zeta_{18}+\zeta_{18}^{2}-\zeta_{18}^{3}+\zeta_{18}^{4}+\cdots)q^{2}+\cdots$
729.2.e.h	$729$	$2$	729.e	27.e	$9$	$6$	$1$	$5.821$	$\Q(\zeta_{18})$	$_{}$	None	None					243.2.a.e	$2$	$0$	$3$	$0$	$-3$	$0$		$1$	$\mathrm{SU}(2)[C_{9}]$	$q+(-\zeta_{18}-\zeta_{18}^{2}+\zeta_{18}^{3}+\zeta_{18}^{4}+\cdots)q^{2}+\cdots$
729.2.e.i	$729$	$2$	729.e	27.e	$9$	$6$	$1$	$5.821$	$\Q(\zeta_{18})$	$_{}$	None	None					243.2.a.e	$2$	$0$	$6$	$0$	$3$	$-9$		$1$	$\mathrm{SU}(2)[C_{9}]$	$q+(1+\zeta_{18}-\zeta_{18}^{4}-\zeta_{18}^{5})q^{2}+(1+\cdots)q^{4}+\cdots$
729.2.e.j	$729$	$2$	729.e	27.e	$9$	$12$	$2$	$5.821$	$\mathbb{Q}[x]/(x^{12} + \cdots)$	$_{}$	None	None		✓			729.2.a.b	$2$	$0$	$-6$	$0$	$6$	$-3$		$3$	$\mathrm{SU}(2)[C_{9}]$	$q+(-1-\beta _{6}-\beta _{7}+\beta _{11})q^{2}+(1-\beta _{1}+\cdots)q^{4}+\cdots$
729.2.e.k	$729$	$2$	729.e	27.e	$9$	$12$	$2$	$5.821$	$\mathbb{Q}[x]/(x^{12} + \cdots)$	$_{}$	None	None		✓			729.2.a.b	$2$	$0$	$-3$	$0$	$-6$	$6$		$3$	$\mathrm{SU}(2)[C_{9}]$	$q+(-\beta _{5}-\beta _{7}+\beta _{10})q^{2}+(-1+\beta _{4}+\cdots)q^{4}+\cdots$
729.2.e.l	$729$	$2$	729.e	27.e	$9$	$12$	$2$	$5.821$	$\mathbb{Q}[x]/(x^{12} + \cdots)$	$_{}$	None	None		✓			729.2.a.b	$2$	$0$	$-3$	$0$	$12$	$-3$		$3$	$\mathrm{SU}(2)[C_{9}]$	$q+\beta _{2}q^{2}+(-2\beta _{1}-\beta _{2}+\beta _{3}-\beta _{5}+\cdots)q^{4}+\cdots$
729.2.e.m	$729$	$2$	729.e	27.e	$9$	$12$	$2$	$5.821$	$\Q(\zeta_{36})$	$_{}$	None	None		✓			729.2.a.c	$4$	$0$	$0$	$0$	$0$	$6$		$3$	$\mathrm{SU}(2)[C_{9}]$	$q+(\beta_{11}-\beta_{9}+\cdots-\beta_{2})q^{2}+\cdots+(-\beta_{5}-1)q^{4}+\cdots$
729.2.e.n	$729$	$2$	729.e	27.e	$9$	$12$	$2$	$5.821$	$\Q(\zeta_{36})$	$_{}$	None	None					243.2.a.c	$12$	$0$	$0$	$0$	$0$	$0$		$3^{3}$	$\mathrm{SU}(2)[C_{9}]$	$q-\beta_{8} q^{2}+(\beta_{7}-\beta_{2})q^{4}-2\beta_{4} q^{5}+\cdots$
729.2.e.o	$729$	$2$	729.e	27.e	$9$	$12$	$2$	$5.821$	$\Q(\zeta_{36})$	$_{}$	None	None					81.2.a.a	$12$	$0$	$0$	$0$	$0$	$0$		$3^{3}$	$\mathrm{SU}(2)[C_{9}]$	$q-\beta_{8} q^{2}+(\beta_{7}-\beta_{2})q^{4}+\beta_{4} q^{5}+\cdots$
729.2.e.p	$729$	$2$	729.e	27.e	$9$	$12$	$2$	$5.821$	12.0. $\cdots$ .1	$_{}$	None	None					243.2.a.d	$12$	$0$	$0$	$0$	$0$	$0$		$3^{3}$	$\mathrm{SU}(2)[C_{9}]$	$q+\beta _{4}q^{2}+4\beta _{5}q^{4}+\beta _{11}q^{5}-2\beta _{7}q^{7}+\cdots$
729.2.e.q	$729$	$2$	729.e	27.e	$9$	$12$	$2$	$5.821$	$\Q(\zeta_{36})$	$_{}$	None	None		✓			729.2.a.c	$4$	$0$	$0$	$0$	$0$	$-12$		$3$	$\mathrm{SU}(2)[C_{9}]$	$q+\beta_{9} q^{2}+(-\beta_{4}+\beta_1+1)q^{4}+\cdots$
729.2.e.r	$729$	$2$	729.e	27.e	$9$	$12$	$2$	$5.821$	$\Q(\zeta_{36})$	$_{}$	None	None		✓			729.2.a.c	$4$	$0$	$0$	$0$	$0$	$6$		$3$	$\mathrm{SU}(2)[C_{9}]$	$q+(\beta_{10}-\beta_{9})q^{2}+(-\beta_{6}-\beta_{4}+\beta_{3}+1)q^{4}+\cdots$
729.2.e.s	$729$	$2$	729.e	27.e	$9$	$12$	$2$	$5.821$	$\mathbb{Q}[x]/(x^{12} + \cdots)$	$_{}$	None	None		✓			729.2.a.b	$2$	$0$	$3$	$0$	$-12$	$-3$		$3$	$\mathrm{SU}(2)[C_{9}]$	$q+(1-\beta _{5}+\beta _{6}+\beta _{10}-\beta _{11})q^{2}+(\beta _{1}+\cdots)q^{4}+\cdots$
729.2.e.t	$729$	$2$	729.e	27.e	$9$	$12$	$2$	$5.821$	$\mathbb{Q}[x]/(x^{12} + \cdots)$	$_{}$	None	None		✓			729.2.a.b	$2$	$0$	$3$	$0$	$6$	$6$		$3$	$\mathrm{SU}(2)[C_{9}]$	$q+(\beta _{1}+\beta _{2}+\beta _{5}-\beta _{6}-\beta _{8})q^{2}+(1+\cdots)q^{4}+\cdots$
729.2.e.u	$729$	$2$	729.e	27.e	$9$	$12$	$2$	$5.821$	$\mathbb{Q}[x]/(x^{12} + \cdots)$	$_{}$	None	None		✓			729.2.a.b	$2$	$0$	$6$	$0$	$-6$	$-3$		$3$	$\mathrm{SU}(2)[C_{9}]$	$q+(\beta _{1}+\beta _{5}-\beta _{6}-\beta _{8})q^{2}+(-1+2\beta _{1}+\cdots)q^{4}+\cdots$
729.2.g.a	$729$	$2$	729.g	81.g	$27$	$144$	$8$	$5.821$		$_{}$	None	None					81.2.g.a	$2$	$0$	$-9$	$0$	$-9$	$9$			$\mathrm{SU}(2)[C_{27}]$
729.2.g.b	$729$	$2$	729.g	81.g	$27$	$144$	$8$	$5.821$		$_{}$	None	None					81.2.g.a	$2$	$0$	$-9$	$0$	$-9$	$9$			$\mathrm{SU}(2)[C_{27}]$
729.2.g.c	$729$	$2$	729.g	81.g	$27$	$144$	$8$	$5.821$		$_{}$	None	None					81.2.g.a	$2$	$0$	$9$	$0$	$9$	$9$			$\mathrm{SU}(2)[C_{27}]$
729.2.g.d	$729$	$2$	729.g	81.g	$27$	$144$	$8$	$5.821$		$_{}$	None	None					81.2.g.a	$2$	$0$	$9$	$0$	$9$	$9$			$\mathrm{SU}(2)[C_{27}]$
729.2.i.a	$729$	$2$	729.i	243.i	$81$	$1404$	$26$	$5.821$		$_{}$	None	None			✓	✓	243.2.i.a	$2$	$0$	$54$	$0$	$54$	$-54$			$\mathrm{SU}(2)[C_{81}]$
729.2.k.a	$729$	$2$	729.k	729.k	$243$	$12960$	$80$	$5.821$		$_{}$	None	None		✓	✓	✓	729.2.k.a	$2$	$0$	$-162$	$-162$	$-162$	$-162$			$\mathrm{SU}(2)[C_{243}]$
729.3.b.a	$729$	$3$	729.b	3.b	$2$	$30$	$30$	$19.864$		$_{}$	None	None					27.3.f.a	$2$	$0$	$0$	$0$	$0$	$0$			$\mathrm{SU}(2)[C_{2}]$
729.3.b.b	$729$	$3$	729.b	3.b	$2$	$36$	$36$	$19.864$		$_{}$	None	None		✓	✓		729.3.b.b	$2$	$0$	$0$	$0$	$0$	$0$			$\mathrm{SU}(2)[C_{2}]$
729.4.a.a	$729$	$4$	729.a	1.a	$1$	$12$	$12$	$43.012$	$\mathbb{Q}[x]/(x^{12} - \cdots)$	$_{}$	None	None	✓	✓			729.4.a.a	$1$	$1$	$-6$	$0$	$-12$	$-42$	$-$	$2^{3}\cdot 3^{5}$	$\mathrm{SU}(2)$	$q+(-1+\beta _{1})q^{2}+(3+\beta _{2}+\beta _{3})q^{4}+\cdots$
729.4.a.b	$729$	$4$	729.a	1.a	$1$	$12$	$12$	$43.012$	$\mathbb{Q}[x]/(x^{12} - \cdots)$	$_{}$	None	None	✓	✓			729.4.a.a	$1$	$1$	$6$	$0$	$12$	$-42$	$-$	$2^{3}\cdot 3^{5}$	$\mathrm{SU}(2)$	$q+(1-\beta _{1})q^{2}+(3+\beta _{2}+\beta _{3})q^{4}+(1+\cdots)q^{5}+\cdots$
729.4.a.c	$729$	$4$	729.a	1.a	$1$	$24$	$24$	$43.012$		$_{}$	None	None	✓		✓		27.4.e.a	$1$	$1$	$-6$	$0$	$-30$	$0$	$-$		$\mathrm{SU}(2)$
729.4.a.d	$729$	$4$	729.a	1.a	$1$	$24$	$24$	$43.012$		$_{}$	None	None	✓				27.4.e.a	$1$	$0$	$6$	$0$	$30$	$0$	$+$		$\mathrm{SU}(2)$
729.4.a.e	$729$	$4$	729.a	1.a	$1$	$30$	$30$	$43.012$		$_{}$	None	None	✓	✓	✓		729.4.a.e	$2$	$0$	$0$	$0$	$0$	$84$	$+$		$\mathrm{SU}(2)$
729.6.a.a	$729$	$6$	729.a	1.a	$1$	$24$	$24$	$116.920$		$_{}$	None	None	✓	✓			729.6.a.a	$1$	$0$	$-12$	$0$	$-33$	$294$	$-$		$\mathrm{SU}(2)$
729.6.a.b	$729$	$6$	729.a	1.a	$1$	$24$	$24$	$116.920$		$_{}$	None	None	✓	✓			729.6.a.a	$1$	$0$	$12$	$0$	$33$	$294$	$-$		$\mathrm{SU}(2)$
729.6.a.c	$729$	$6$	729.a	1.a	$1$	$42$	$42$	$116.920$		$_{}$	None	None	✓				27.6.e.a	$1$	$1$	$-12$	$0$	$-150$	$0$	$+$		$\mathrm{SU}(2)$
729.6.a.d	$729$	$6$	729.a	1.a	$1$	$42$	$42$	$116.920$		$_{}$	None	None	✓	✓			729.6.a.d	$2$	$1$	$0$	$0$	$0$	$-588$	$+$		$\mathrm{SU}(2)$
729.6.a.e	$729$	$6$	729.a	1.a	$1$	$42$	$42$	$116.920$		$_{}$	None	None	✓		✓		27.6.e.a	$1$	$0$	$12$	$0$	$150$	$0$	$-$		$\mathrm{SU}(2)$

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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.