Elliptic curve isogeny class with LMFDB label 540.d (Cremona label 540b)

• Label: 540.d
• Number of curves: $2$
• Conductor: $540$
• CM: no
• Rank: $1$

Rank

The elliptic curves in class 540.d have rank \(1\).

L-function data

Bad L-factors:

Prime	L-Factor
\(2\)	\(1\)
\(3\)	\(1\)
\(5\)	\(1 - T\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(7\)	\( 1 + 4 T + 7 T^{2}\)	1.7.e
\(11\)	\( 1 + 6 T + 11 T^{2}\)	1.11.g
\(13\)	\( 1 + 4 T + 13 T^{2}\)	1.13.e
\(17\)	\( 1 - 3 T + 17 T^{2}\)	1.17.ad
\(19\)	\( 1 + 7 T + 19 T^{2}\)	1.19.h
\(23\)	\( 1 - 9 T + 23 T^{2}\)	1.23.aj
\(29\)	\( 1 + 29 T^{2}\)	1.29.a
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 540.d do not have complex multiplication.

Modular form 540.2.a.d

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.

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

Isogeny graph

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

Elliptic curves in class 540.d

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
540.d1	540b2	\([0, 0, 0, -57, 169]\)	\(-5267712/125\)	\(-486000\)	\([3]\)	\(108\)	\(-0.12251\)
540.d2	540b1	\([0, 0, 0, 3, 1]\)	\(6912/5\)	\(-2160\)	\([]\)	\(36\)	\(-0.67182\)	\(\Gamma_0(N)\)-optimal