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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 (25 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				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*	$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$	Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
538.2.a.a	$538$	$2$	538.a	1.a	$1$	$2$	$2$	$4.296$	$\Q(\sqrt{5}) $	None	✓	✓	✓	✓	538.2.a.a	$1$	$1$	$2$	$-1$	$-5$	$-4$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-\beta q^{3}+q^{4}+(-3+\beta )q^{5}-\beta q^{6}+\cdots$
538.2.a.b	$538$	$2$	538.a	1.a	$1$	$2$	$2$	$4.296$	$\Q(\sqrt{13}) $	None	✓	✓			538.2.a.b	$1$	$0$	$2$	$1$	$1$	$-2$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+\beta q^{3}+q^{4}+(1-\beta )q^{5}+\beta q^{6}+\cdots$
538.2.a.c	$538$	$2$	538.a	1.a	$1$	$4$	$4$	$4.296$	4.4.4913.1	None	✓	✓	✓	✓	538.2.a.c	$1$	$0$	$-4$	$3$	$5$	$-1$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+(1-\beta _{1})q^{3}+q^{4}+(2-\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots$
538.2.a.d	$538$	$2$	538.a	1.a	$1$	$7$	$7$	$4.296$	$\mathbb{Q}[x]/(x^{7} - \cdots)$	None	✓	✓	✓	✓	538.2.a.d	$1$	$1$	$-7$	$-4$	$-6$	$-3$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+(-1+\beta _{1}+\beta _{6})q^{3}+q^{4}+(\beta _{3}+\cdots)q^{5}+\cdots$
538.2.a.e	$538$	$2$	538.a	1.a	$1$	$7$	$7$	$4.296$	$\mathbb{Q}[x]/(x^{7} - \cdots)$	None	✓	✓	✓		538.2.a.e	$1$	$0$	$7$	$1$	$7$	$6$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+\beta _{2}q^{3}+q^{4}+(1+\beta _{4})q^{5}+\beta _{2}q^{6}+\cdots$
538.2.b.a	$538$	$2$	538.b	269.b	$2$	$4$	$4$	$4.296$	$\Q(i, \sqrt{5})$	None		✓			538.2.b.a	$2$	$0$	$0$	$0$	$6$	$0$		$1$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta _{3}q^{2}+\beta _{1}q^{3}-q^{4}+(1-\beta _{2})q^{5}+\cdots$
538.2.b.b	$538$	$2$	538.b	269.b	$2$	$18$	$18$	$4.296$	$\mathbb{Q}[x]/(x^{18} + \cdots)$	None		✓	✓		538.2.b.b	$2$	$0$	$0$	$0$	$-4$	$0$		$1$	$\mathrm{SU}(2)[C_{2}]$	$q-\beta _{8}q^{2}-\beta _{1}q^{3}-q^{4}-\beta _{11}q^{5}-\beta _{3}q^{6}+\cdots$
538.2.d.a	$538$	$2$	538.d	269.d	$67$	$726$	$11$	$4.296$		None		✓			538.2.d.a	$2$	$0$	$11$	$1$	$1$	$4$			$\mathrm{SU}(2)[C_{67}]$
538.2.d.b	$538$	$2$	538.d	269.d	$67$	$792$	$12$	$4.296$		None		✓	✓		538.2.d.b	$2$	$0$	$-12$	$-5$	$-9$	$-8$			$\mathrm{SU}(2)[C_{67}]$
538.2.e.a	$538$	$2$	538.e	269.e	$134$	$1452$	$22$	$4.296$		None		✓	✓	✓	538.2.e.a	$2$	$0$	$0$	$0$	$-2$	$0$			$\mathrm{SU}(2)[C_{134}]$
538.3.c.a	$538$	$3$	538.c	269.c	$4$	$44$	$22$	$14.659$		None		✓			538.3.c.a	$2$	$0$	$44$	$2$	$0$	$4$			$\mathrm{SU}(2)[C_{4}]$
538.3.c.b	$538$	$3$	538.c	269.c	$4$	$46$	$23$	$14.659$		None		✓	✓		538.3.c.b	$2$	$0$	$-46$	$-6$	$-32$	$4$			$\mathrm{SU}(2)[C_{4}]$
538.4.a.a	$538$	$4$	538.a	1.a	$1$	$13$	$13$	$31.743$	$\mathbb{Q}[x]/(x^{13} - \cdots)$	None	✓	✓	✓	✓	538.4.a.a	$1$	$1$	$26$	$-15$	$-41$	$-60$	$-$	$1$	$\mathrm{SU}(2)$	$q+2q^{2}+(-1-\beta _{1})q^{3}+4q^{4}+(-3+\cdots)q^{5}+\cdots$
538.4.a.b	$538$	$4$	538.a	1.a	$1$	$15$	$15$	$31.743$	$\mathbb{Q}[x]/(x^{15} - \cdots)$	None	✓	✓	✓	✓	538.4.a.b	$1$	$0$	$-30$	$11$	$29$	$5$	$+$	$1$	$\mathrm{SU}(2)$	$q-2q^{2}+(1-\beta _{1})q^{3}+4q^{4}+(2+\beta _{4}+\cdots)q^{5}+\cdots$
538.4.a.c	$538$	$4$	538.a	1.a	$1$	$18$	$18$	$31.743$	$\mathbb{Q}[x]/(x^{18} - \cdots)$	None	✓	✓	✓	✓	538.4.a.c	$1$	$1$	$-36$	$-10$	$-26$	$-9$	$-$	$3^{2}$	$\mathrm{SU}(2)$	$q-2q^{2}+(-1+\beta _{1})q^{3}+4q^{4}+(-1+\cdots)q^{5}+\cdots$
538.4.a.d	$538$	$4$	538.a	1.a	$1$	$21$	$21$	$31.743$		None	✓	✓	✓	✓	538.4.a.d	$1$	$0$	$42$	$6$	$54$	$52$	$+$		$\mathrm{SU}(2)$
538.4.b.a	$538$	$4$	538.b	269.b	$2$	$68$	$68$	$31.743$		None		✓	✓	✓	538.4.b.a	$2$	$0$	$0$	$0$	$38$	$0$			$\mathrm{SU}(2)[C_{2}]$
538.6.a.a	$538$	$6$	538.a	1.a	$1$	$24$	$24$	$86.286$		None	✓	✓	✓	✓	538.6.a.a	$1$	$1$	$96$	$-31$	$-261$	$-356$	$+$		$\mathrm{SU}(2)$
538.6.a.b	$538$	$6$	538.a	1.a	$1$	$27$	$27$	$86.286$		None	✓	✓	✓	✓	538.6.a.b	$1$	$0$	$-108$	$33$	$139$	$-25$	$-$		$\mathrm{SU}(2)$
538.6.a.c	$538$	$6$	538.a	1.a	$1$	$30$	$30$	$86.286$		None	✓	✓	✓	✓	538.6.a.c	$1$	$1$	$-120$	$-30$	$-136$	$-123$	$+$		$\mathrm{SU}(2)$
538.6.a.d	$538$	$6$	538.a	1.a	$1$	$32$	$32$	$86.286$		None	✓	✓	✓	✓	538.6.a.d	$1$	$0$	$128$	$32$	$214$	$428$	$-$		$\mathrm{SU}(2)$
538.8.a.a	$538$	$8$	538.a	1.a	$1$	$35$	$35$	$168.063$		None	✓	✓	✓	✓	538.8.a.a	$1$	$1$	$280$	$-66$	$-1126$	$-3196$	$-$		$\mathrm{SU}(2)$
538.8.a.b	$538$	$8$	538.a	1.a	$1$	$37$	$37$	$168.063$		None	✓	✓	✓	✓	538.8.a.b	$1$	$0$	$-296$	$80$	$624$	$453$	$+$		$\mathrm{SU}(2)$
538.8.a.c	$538$	$8$	538.a	1.a	$1$	$40$	$40$	$168.063$		None	✓	✓	✓	✓	538.8.a.c	$1$	$1$	$-320$	$-109$	$-751$	$-233$	$-$		$\mathrm{SU}(2)$
538.8.a.d	$538$	$8$	538.a	1.a	$1$	$43$	$43$	$168.063$		None	✓	✓	✓	✓	538.8.a.d	$1$	$0$	$344$	$123$	$1249$	$2292$	$+$		$\mathrm{SU}(2)$

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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.