Elliptic curve isogeny class with Cremona label 1225a (LMFDB label 1225.e)

• Label: 1225a
• Number of curves: $3$
• Conductor: $1225$
• CM: no
• Rank: $0$

Rank

The elliptic curves in class 1225a have rank \(0\).

L-function data

Bad L-factors:

Prime	L-Factor
\(5\)	\(1\)
\(7\)	\(1\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(2\)	\( 1 + 2 T + 2 T^{2}\)	1.2.c
\(3\)	\( 1 - T + 3 T^{2}\)	1.3.ab
\(11\)	\( 1 + 3 T + 11 T^{2}\)	1.11.d
\(13\)	\( 1 - T + 13 T^{2}\)	1.13.ab
\(17\)	\( 1 - 7 T + 17 T^{2}\)	1.17.ah
\(19\)	\( 1 + 19 T^{2}\)	1.19.a
\(23\)	\( 1 + 6 T + 23 T^{2}\)	1.23.g
\(29\)	\( 1 + 5 T + 29 T^{2}\)	1.29.f
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 1225a do not have complex multiplication.

Modular form 1225.2.a.a

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.

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

Isogeny graph

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

Elliptic curves in class 1225a

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
1225.e2	1225a1	\([0, 1, 1, -1633, -28731]\)	\(-262144/35\)	\(-64339296875\)	\([]\)	\(768\)	\(0.80652\)	\(\Gamma_0(N)\)-optimal
1225.e3	1225a2	\([0, 1, 1, 10617, 75394]\)	\(71991296/42875\)	\(-78815638671875\)	\([]\)	\(2304\)	\(1.3558\)
1225.e1	1225a3	\([0, 1, 1, -160883, 25929019]\)	\(-250523582464/13671875\)	\(-25132537841796875\)	\([]\)	\(6912\)	\(1.9051\)