Elliptic curve isogeny class with Cremona label 130b (LMFDB label 130.b)

• Label: 130b
• Number of curves: $4$
• Conductor: $130$
• CM: no
• Rank: $0$

Rank

The elliptic curves in class 130b have rank \(0\).

L-function data

Bad L-factors:

Prime	L-Factor
\(2\)	\(1 - T\)
\(5\)	\(1 - T\)
\(13\)	\(1 - T\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(3\)	\( 1 + 3 T^{2}\)	1.3.a
\(7\)	\( 1 + 7 T^{2}\)	1.7.a
\(11\)	\( 1 + 11 T^{2}\)	1.11.a
\(17\)	\( 1 - 2 T + 17 T^{2}\)	1.17.ac
\(19\)	\( 1 + 8 T + 19 T^{2}\)	1.19.i
\(23\)	\( 1 + 4 T + 23 T^{2}\)	1.23.e
\(29\)	\( 1 + 2 T + 29 T^{2}\)	1.29.c
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 130b do not have complex multiplication.

Modular form 130.2.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 Cremona numbering.

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

Elliptic curves in class 130b

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
130.b4	130b1	\([1, -1, 1, -7, -1]\)	\(33076161/16640\)	\(16640\)	\([4]\)	\(8\)	\(-0.49736\)	\(\Gamma_0(N)\)-optimal
130.b2	130b2	\([1, -1, 1, -87, -289]\)	\(72043225281/67600\)	\(67600\)	\([2, 2]\)	\(16\)	\(-0.15079\)
130.b1	130b3	\([1, -1, 1, -1387, -19529]\)	\(294889639316481/260\)	\(260\)	\([2]\)	\(32\)	\(0.19579\)
130.b3	130b4	\([1, -1, 1, -67, -441]\)	\(-32798729601/71402500\)	\(-71402500\)	\([4]\)	\(32\)	\(0.19579\)