Elliptic curve isogeny class with Cremona label 2040f (LMFDB label 2040.h)

• Label: 2040f
• Number of curves: $4$
• Conductor: $2040$
• CM: no
• Rank: $0$

Rank

The elliptic curves in class 2040f have rank \(0\).

L-function data

Bad L-factors:

Prime	L-Factor
\(2\)	\(1\)
\(3\)	\(1 + T\)
\(5\)	\(1 - T\)
\(17\)	\(1 - T\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(7\)	\( 1 + 7 T^{2}\)	1.7.a
\(11\)	\( 1 + 11 T^{2}\)	1.11.a
\(13\)	\( 1 + 2 T + 13 T^{2}\)	1.13.c
\(19\)	\( 1 + 4 T + 19 T^{2}\)	1.19.e
\(23\)	\( 1 + 8 T + 23 T^{2}\)	1.23.i
\(29\)	\( 1 + 2 T + 29 T^{2}\)	1.29.c
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 2040f do not have complex multiplication.

Modular form 2040.2.a.f

Isogeny matrix

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

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

The vertices are labelled with Cremona labels.

Elliptic curves in class 2040f

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
2040.h4	2040f1	\([0, 1, 0, -20236, 33628064]\)	\(-3579968623693264/1906997690433375\)	\(-488191408750944000\)	\([2]\)	\(26880\)	\(2.0732\)	\(\Gamma_0(N)\)-optimal
2040.h3	2040f2	\([0, 1, 0, -1690656, 836766000]\)	\(521902963282042184836/6241849278890625\)	\(6391653661584000000\)	\([2, 2]\)	\(53760\)	\(2.4198\)
2040.h2	2040f3	\([0, 1, 0, -3135656, -807066000]\)	\(1664865424893526702418/826424127435466125\)	\(1692516612987834624000\)	\([2]\)	\(107520\)	\(2.7664\)
2040.h1	2040f4	\([0, 1, 0, -26972376, 53908152624]\)	\(1059623036730633329075378/154307373046875\)	\(316021500000000000\)	\([2]\)	\(107520\)	\(2.7664\)