Elliptic curve isogeny class with Cremona label 672f (LMFDB label 672.f)

• Label: 672f
• Number of curves: $4$
• Conductor: $672$
• CM: no
• Rank: $1$

Rank

The elliptic curves in class 672f have rank \(1\).

L-function data

Bad L-factors:

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

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(5\)	\( 1 + 2 T + 5 T^{2}\)	1.5.c
\(11\)	\( 1 + 4 T + 11 T^{2}\)	1.11.e
\(13\)	\( 1 + 6 T + 13 T^{2}\)	1.13.g
\(17\)	\( 1 + 2 T + 17 T^{2}\)	1.17.c
\(19\)	\( 1 - 4 T + 19 T^{2}\)	1.19.ae
\(23\)	\( 1 + 4 T + 23 T^{2}\)	1.23.e
\(29\)	\( 1 + 2 T + 29 T^{2}\)	1.29.c
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 672f do not have complex multiplication.

Modular form 672.2.a.f

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 Cremona numbering.

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

Isogeny graph

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

Elliptic curves in class 672f

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
672.f3	672f1	\([0, 1, 0, -14, -24]\)	\(5088448/441\)	\(28224\)	\([2, 2]\)	\(64\)	\(-0.40521\)	\(\Gamma_0(N)\)-optimal
672.f1	672f2	\([0, 1, 0, -224, -1368]\)	\(2438569736/21\)	\(10752\)	\([2]\)	\(128\)	\(-0.058633\)
672.f2	672f3	\([0, 1, 0, -49, 95]\)	\(3241792/567\)	\(2322432\)	\([4]\)	\(128\)	\(-0.058633\)
672.f4	672f4	\([0, 1, 0, 16, -84]\)	\(830584/7203\)	\(-3687936\)	\([2]\)	\(128\)	\(-0.058633\)