Elliptic curve isogeny class with LMFDB label 3648.w (Cremona label 3648bg)

• Label: 3648.w
• Number of curves: $4$
• Conductor: $3648$
• CM: no
• Rank: $1$

Rank

The elliptic curves in class 3648.w have rank \(1\).

L-function data

Bad L-factors:

Prime	L-Factor
\(2\)	\(1\)
\(3\)	\(1 - T\)
\(19\)	\(1 + T\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(5\)	\( 1 + 2 T + 5 T^{2}\)	1.5.c
\(7\)	\( 1 + 7 T^{2}\)	1.7.a
\(11\)	\( 1 + 11 T^{2}\)	1.11.a
\(13\)	\( 1 + 2 T + 13 T^{2}\)	1.13.c
\(17\)	\( 1 - 2 T + 17 T^{2}\)	1.17.ac
\(23\)	\( 1 + 23 T^{2}\)	1.23.a
\(29\)	\( 1 + 2 T + 29 T^{2}\)	1.29.c
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 3648.w do not have complex multiplication.

Modular form 3648.2.a.w

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 LMFDB 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 LMFDB labels.

Elliptic curves in class 3648.w

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
3648.w1	3648bg3	\([0, 1, 0, -5089, -137569]\)	\(111223479026/3518667\)	\(461198721024\)	\([2]\)	\(3072\)	\(1.0128\)
3648.w2	3648bg2	\([0, 1, 0, -769, 4991]\)	\(768400132/263169\)	\(17247043584\)	\([2, 2]\)	\(1536\)	\(0.66620\)
3648.w3	3648bg1	\([0, 1, 0, -689, 6735]\)	\(2211014608/513\)	\(8404992\)	\([2]\)	\(768\)	\(0.31963\)	\(\Gamma_0(N)\)-optimal
3648.w4	3648bg4	\([0, 1, 0, 2271, 37215]\)	\(9878111854/10097379\)	\(-1323483660288\)	\([4]\)	\(3072\)	\(1.0128\)