Elliptic curve isogeny class with LMFDB label 725.a (Cremona label 725a)

• Label: 725.a
• Number of curves: $2$
• Conductor: $725$
• CM: no
• Rank: $1$

Rank

The elliptic curves in class 725.a have rank \(1\).

L-function data

Bad L-factors:

Prime	L-Factor
\(5\)	\(1\)
\(29\)	\(1 + T\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(2\)	\( 1 - T + 2 T^{2}\)	1.2.ab
\(3\)	\( 1 + 3 T^{2}\)	1.3.a
\(7\)	\( 1 - 2 T + 7 T^{2}\)	1.7.ac
\(11\)	\( 1 + 6 T + 11 T^{2}\)	1.11.g
\(13\)	\( 1 + 2 T + 13 T^{2}\)	1.13.c
\(17\)	\( 1 - 2 T + 17 T^{2}\)	1.17.ac
\(19\)	\( 1 + 2 T + 19 T^{2}\)	1.19.c
\(23\)	\( 1 + 2 T + 23 T^{2}\)	1.23.c
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 725.a do not have complex multiplication.

Modular form 725.2.a.a

Isogeny matrix

The \((i,j)\)-th entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.

\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)

Isogeny graph

The vertices are labelled with LMFDB labels, and the \( \Gamma_0(N) \)-optimal curve is highlighted in blue.

Elliptic curves in class 725.a

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
725.a1	725a1	\([1, -1, 0, -67, 216]\)	\(2146689/145\)	\(2265625\)	\([2]\)	\(96\)	\(-0.031782\)	\(\Gamma_0(N)\)-optimal
725.a2	725a2	\([1, -1, 0, 58, 841]\)	\(1367631/21025\)	\(-328515625\)	\([2]\)	\(192\)	\(0.31479\)