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

• Label: 672c
• Number of curves: $4$
• Conductor: $672$
• CM: no
• Rank: $0$

Rank

The elliptic curves in class 672c have rank \(0\).

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

See L-function page for more information

Complex multiplication

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

Modular form 672.2.a.c

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 672c

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
672.d3	672c1	\([0, -1, 0, -22, 40]\)	\(19248832/3969\)	\(254016\)	\([2, 2]\)	\(64\)	\(-0.24763\)	\(\Gamma_0(N)\)-optimal
672.d2	672c2	\([0, -1, 0, -112, -392]\)	\(306182024/21609\)	\(11063808\)	\([2]\)	\(128\)	\(0.098947\)
672.d1	672c3	\([0, -1, 0, -337, 2497]\)	\(1036433728/63\)	\(258048\)	\([4]\)	\(128\)	\(0.098947\)
672.d4	672c4	\([0, -1, 0, 48, 180]\)	\(23393656/45927\)	\(-23514624\)	\([2]\)	\(128\)	\(0.098947\)