Elliptic curve isogeny class with LMFDB label 50.a (Cremona label 50a)

• Label: 50.a
• Number of curves: $4$
• Conductor: $50$
• CM: no
• Rank: $0$

Rank

The elliptic curves in class 50.a have rank \(0\).

L-function data

Bad L-factors:

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

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(3\)	\( 1 - T + 3 T^{2}\)	1.3.ab
\(7\)	\( 1 - 2 T + 7 T^{2}\)	1.7.ac
\(11\)	\( 1 + 3 T + 11 T^{2}\)	1.11.d
\(13\)	\( 1 + 4 T + 13 T^{2}\)	1.13.e
\(17\)	\( 1 + 3 T + 17 T^{2}\)	1.17.d
\(19\)	\( 1 - 5 T + 19 T^{2}\)	1.19.af
\(23\)	\( 1 - 6 T + 23 T^{2}\)	1.23.ag
\(29\)	\( 1 + 29 T^{2}\)	1.29.a
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 50.a do not have complex multiplication.

Modular form 50.2.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 LMFDB numbering.

\(\left(\begin{array}{rrrr} 1 & 15 & 3 & 5 \\ 15 & 1 & 5 & 3 \\ 3 & 5 & 1 & 15 \\ 5 & 3 & 15 & 1 \end{array}\right)\)

Isogeny graph

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

Elliptic curves in class 50.a

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
50.a1	50a2	\([1, 0, 1, -126, -552]\)	\(-349938025/8\)	\(-5000\)	\([]\)	\(6\)	\(-0.17679\)
50.a2	50a3	\([1, 0, 1, -76, 298]\)	\(-121945/32\)	\(-12500000\)	\([3]\)	\(10\)	\(0.078619\)
50.a3	50a1	\([1, 0, 1, -1, -2]\)	\(-25/2\)	\(-1250\)	\([3]\)	\(2\)	\(-0.72610\)	\(\Gamma_0(N)\)-optimal
50.a4	50a4	\([1, 0, 1, 549, -2202]\)	\(46969655/32768\)	\(-12800000000\)	\([]\)	\(30\)	\(0.62793\)