Elliptic curve isogeny class with Cremona label 51205o (LMFDB label 51205.d)

• Label: 51205o
• Number of curves: $2$
• Conductor: $51205$
• CM: no
• Rank: $0$

Rank

The elliptic curves in class 51205o have rank \(0\).

L-function data

Bad L-factors:

Prime	L-Factor
\(5\)	\(1 + T\)
\(7\)	\(1\)
\(11\)	\(1 + T\)
\(19\)	\(1 - T\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(2\)	\( 1 - T + 2 T^{2}\)	1.2.ab
\(3\)	\( 1 + 2 T + 3 T^{2}\)	1.3.c
\(13\)	\( 1 - 4 T + 13 T^{2}\)	1.13.ae
\(17\)	\( 1 + 6 T + 17 T^{2}\)	1.17.g
\(23\)	\( 1 + 2 T + 23 T^{2}\)	1.23.c
\(29\)	\( 1 + 4 T + 29 T^{2}\)	1.29.e
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 51205o do not have complex multiplication.

Modular form 51205.2.a.o

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 51205o

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
51205.d2	51205o1	\([1, 1, 1, -295, 1420]\)	\(24137569/5225\)	\(614716025\)	\([2]\)	\(23040\)	\(0.40034\)	\(\Gamma_0(N)\)-optimal
51205.d1	51205o2	\([1, 1, 1, -1520, -22100]\)	\(3301293169/218405\)	\(25695129845\)	\([2]\)	\(46080\)	\(0.74691\)