Elliptic curve isogeny class with LMFDB label 2205.d (Cremona label 2205f)

• Label: 2205.d
• Number of curves: $2$
• Conductor: $2205$
• CM: no
• Rank: $1$

Rank

The elliptic curves in class 2205.d have rank \(1\).

L-function data

Bad L-factors:

Prime	L-Factor
\(3\)	\(1\)
\(5\)	\(1 + T\)
\(7\)	\(1\)

Good L-factors:

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

See L-function page for more information

Complex multiplication

The elliptic curves in class 2205.d do not have complex multiplication.

Modular form 2205.2.a.d

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.

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

Isogeny graph

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

Elliptic curves in class 2205.d

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
2205.d1	2205f2	\([0, 0, 1, -1848, -30641]\)	\(-19539165184/46875\)	\(-1674421875\)	\([]\)	\(1152\)	\(0.64922\)
2205.d2	2205f1	\([0, 0, 1, 42, -212]\)	\(229376/675\)	\(-24111675\)	\([]\)	\(384\)	\(0.099916\)	\(\Gamma_0(N)\)-optimal