Elliptic curve isogeny class with Cremona label 1005a (LMFDB label 1005.b)

• Label: 1005a
• Number of curves: $2$
• Conductor: $1005$
• CM: no
• Rank: $0$

Rank

The elliptic curves in class 1005a have rank \(0\).

L-function data

Bad L-factors:

Prime	L-Factor
\(3\)	\(1 - T\)
\(5\)	\(1 + T\)
\(67\)	\(1 - T\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(2\)	\( 1 + 2 T^{2}\)	1.2.a
\(7\)	\( 1 - 2 T + 7 T^{2}\)	1.7.ac
\(11\)	\( 1 + 6 T + 11 T^{2}\)	1.11.g
\(13\)	\( 1 - 2 T + 13 T^{2}\)	1.13.ac
\(17\)	\( 1 + 3 T + 17 T^{2}\)	1.17.d
\(19\)	\( 1 + T + 19 T^{2}\)	1.19.b
\(23\)	\( 1 + 9 T + 23 T^{2}\)	1.23.j
\(29\)	\( 1 - 3 T + 29 T^{2}\)	1.29.ad
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 1005a do not have complex multiplication.

Modular form 1005.2.a.a

Isogeny matrix

The \(i,j\) 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.

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

Isogeny graph

The vertices are labelled with Cremona labels.

Elliptic curves in class 1005a

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
1005.b1	1005a1	\([1, 1, 0, -297, -1944]\)	\(2912566550041/254390625\)	\(254390625\)	\([2]\)	\(360\)	\(0.35342\)	\(\Gamma_0(N)\)-optimal
1005.b2	1005a2	\([1, 1, 0, 328, -8319]\)	\(3883959939959/33133870125\)	\(-33133870125\)	\([2]\)	\(720\)	\(0.69999\)