Elliptic curve isogeny class with LMFDB label 475.b (Cremona label 475a)

• Label: 475.b
• Number of curves: $3$
• Conductor: $475$
• CM: no
• Rank: $0$

Rank

The elliptic curves in class 475.b have rank \(0\).

L-function data

Bad L-factors:

Prime	L-Factor
\(5\)	\(1\)
\(19\)	\(1 - T\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(2\)	\( 1 + 2 T^{2}\)	1.2.a
\(3\)	\( 1 - 2 T + 3 T^{2}\)	1.3.ac
\(7\)	\( 1 - T + 7 T^{2}\)	1.7.ab
\(11\)	\( 1 - 3 T + 11 T^{2}\)	1.11.ad
\(13\)	\( 1 - 4 T + 13 T^{2}\)	1.13.ae
\(17\)	\( 1 - 3 T + 17 T^{2}\)	1.17.ad
\(23\)	\( 1 + 23 T^{2}\)	1.23.a
\(29\)	\( 1 - 6 T + 29 T^{2}\)	1.29.ag
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 475.b do not have complex multiplication.

Modular form 475.2.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.

\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

Elliptic curves in class 475.b

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
475.b1	475a3	\([0, -1, 1, -19233, -1020257]\)	\(-50357871050752/19\)	\(-296875\)	\([]\)	\(324\)	\(0.83816\)
475.b2	475a2	\([0, -1, 1, -233, -1382]\)	\(-89915392/6859\)	\(-107171875\)	\([]\)	\(108\)	\(0.28885\)
475.b3	475a1	\([0, -1, 1, 17, -7]\)	\(32768/19\)	\(-296875\)	\([]\)	\(36\)	\(-0.26045\)	\(\Gamma_0(N)\)-optimal