Elliptic curve isogeny class with LMFDB label 1305.e (Cremona label 1305c)

• Label: 1305.e
• Number of curves: $4$
• Conductor: $1305$
• CM: no
• Rank: $0$

Rank

The elliptic curves in class 1305.e have rank \(0\).

L-function data

Bad L-factors:

Prime	L-Factor
\(3\)	\(1\)
\(5\)	\(1 + T\)
\(29\)	\(1 + T\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(2\)	\( 1 - T + 2 T^{2}\)	1.2.ab
\(7\)	\( 1 + 4 T + 7 T^{2}\)	1.7.e
\(11\)	\( 1 + 11 T^{2}\)	1.11.a
\(13\)	\( 1 - 6 T + 13 T^{2}\)	1.13.ag
\(17\)	\( 1 + 2 T + 17 T^{2}\)	1.17.c
\(19\)	\( 1 - 8 T + 19 T^{2}\)	1.19.ai
\(23\)	\( 1 - 4 T + 23 T^{2}\)	1.23.ae
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 1305.e do not have complex multiplication.

Modular form 1305.2.a.e

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.

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

Isogeny graph

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

Elliptic curves in class 1305.e

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
1305.e1	1305c3	\([1, -1, 0, -62640, 6049971]\)	\(37286818682653441/1305\)	\(951345\)	\([2]\)	\(2560\)	\(1.0934\)
1305.e2	1305c2	\([1, -1, 0, -3915, 95256]\)	\(9104453457841/1703025\)	\(1241505225\)	\([2, 2]\)	\(1280\)	\(0.74685\)
1305.e3	1305c4	\([1, -1, 0, -3510, 115425]\)	\(-6561258219361/3978455625\)	\(-2900294150625\)	\([2]\)	\(2560\)	\(1.0934\)
1305.e4	1305c1	\([1, -1, 0, -270, 1215]\)	\(2992209121/951345\)	\(693530505\)	\([2]\)	\(640\)	\(0.40028\)	\(\Gamma_0(N)\)-optimal