Elliptic curve isogeny class with Cremona label 49a (LMFDB label 49.a)

• Label: 49a
• Number of curves: $4$
• Conductor: $49$
• CM: \(\Q(\sqrt{-7}) \)
• Rank: $0$

Rank

The elliptic curves in class 49a have rank \(0\).

L-function data

Bad L-factors:

Prime	L-Factor
\(7\)	\(1\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(2\)	\( 1 - T + 2 T^{2}\)	1.2.ab
\(3\)	\( 1 + 3 T^{2}\)	1.3.a
\(5\)	\( 1 + 5 T^{2}\)	1.5.a
\(11\)	\( 1 - 4 T + 11 T^{2}\)	1.11.ae
\(13\)	\( 1 + 13 T^{2}\)	1.13.a
\(17\)	\( 1 + 17 T^{2}\)	1.17.a
\(19\)	\( 1 + 19 T^{2}\)	1.19.a
\(23\)	\( 1 - 8 T + 23 T^{2}\)	1.23.ai
\(29\)	\( 1 - 2 T + 29 T^{2}\)	1.29.ac
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

Each elliptic curve in class 49a has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-7}) \).

Modular form 49.2.a.a

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.

\(\left(\begin{array}{rrrr} 1 & 2 & 7 & 14 \\ 2 & 1 & 14 & 7 \\ 7 & 14 & 1 & 2 \\ 14 & 7 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

Elliptic curves in class 49a

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality	CM discriminant
49.a4	49a1	\([1, -1, 0, -2, -1]\)	\(-3375\)	\(-343\)	\([2]\)	\(1\)	\(-0.79914\)	\(\Gamma_0(N)\)-optimal	\(-7\)
49.a3	49a2	\([1, -1, 0, -37, -78]\)	\(16581375\)	\(343\)	\([2]\)	\(2\)	\(-0.45256\)		\(-28\)
49.a2	49a3	\([1, -1, 0, -107, 552]\)	\(-3375\)	\(-40353607\)	\([2]\)	\(7\)	\(0.17382\)		\(-7\)
49.a1	49a4	\([1, -1, 0, -1822, 30393]\)	\(16581375\)	\(40353607\)	\([2]\)	\(14\)	\(0.52039\)		\(-28\)