Elliptic curve isogeny class with LMFDB label 5040.r (Cremona label 5040bh)

• Label: 5040.r
• Number of curves: $2$
• Conductor: $5040$
• CM: no
• Rank: $0$

Rank

The elliptic curves in class 5040.r have rank \(0\).

L-function data

Bad L-factors:

Prime	L-Factor
\(2\)	\(1\)
\(3\)	\(1\)
\(5\)	\(1 + T\)
\(7\)	\(1 - T\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(11\)	\( 1 - 2 T + 11 T^{2}\)	1.11.ac
\(13\)	\( 1 - 4 T + 13 T^{2}\)	1.13.ae
\(17\)	\( 1 + 2 T + 17 T^{2}\)	1.17.c
\(19\)	\( 1 - 2 T + 19 T^{2}\)	1.19.ac
\(23\)	\( 1 - 4 T + 23 T^{2}\)	1.23.ae
\(29\)	\( 1 - 2 T + 29 T^{2}\)	1.29.ac
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 5040.r do not have complex multiplication.

Modular form 5040.2.a.r

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 5040.r

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
5040.r1	5040bh1	\([0, 0, 0, -48, 47]\)	\(1048576/525\)	\(6123600\)	\([2]\)	\(768\)	\(-0.0049075\)	\(\Gamma_0(N)\)-optimal
5040.r2	5040bh2	\([0, 0, 0, 177, 362]\)	\(3286064/2205\)	\(-411505920\)	\([2]\)	\(1536\)	\(0.34167\)