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

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

Rank

The elliptic curves in class 50.b 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.b
\(7\)	\( 1 + 2 T + 7 T^{2}\)	1.7.c
\(11\)	\( 1 + 3 T + 11 T^{2}\)	1.11.d
\(13\)	\( 1 - 4 T + 13 T^{2}\)	1.13.ae
\(17\)	\( 1 - 3 T + 17 T^{2}\)	1.17.ad
\(19\)	\( 1 - 5 T + 19 T^{2}\)	1.19.af
\(23\)	\( 1 + 6 T + 23 T^{2}\)	1.23.g
\(29\)	\( 1 + 29 T^{2}\)	1.29.a
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

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

Modular form 50.2.a.b

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.

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

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.b

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
50.b1	50b4	\([1, 1, 1, -3138, -68969]\)	\(-349938025/8\)	\(-78125000\)	\([]\)	\(30\)	\(0.62793\)
50.b2	50b3	\([1, 1, 1, -13, -219]\)	\(-25/2\)	\(-19531250\)	\([]\)	\(10\)	\(0.078619\)
50.b3	50b1	\([1, 1, 1, -3, 1]\)	\(-121945/32\)	\(-800\)	\([5]\)	\(2\)	\(-0.72610\)	\(\Gamma_0(N)\)-optimal
50.b4	50b2	\([1, 1, 1, 22, -9]\)	\(46969655/32768\)	\(-819200\)	\([5]\)	\(6\)	\(-0.17679\)