# Properties

 Label 2205.d Number of curves $2$ Conductor $2205$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("d1")

sage: E.isogeny_class()

## Elliptic curves in class 2205.d

sage: E.isogeny_class().curves

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

## Rank

sage: E.rank()

The elliptic curves in class 2205.d have rank $$1$$.

## Complex multiplication

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

## Modular form2205.2.a.d

sage: E.q_eigenform(10)

$$q - 2q^{4} - q^{5} + q^{13} + 4q^{16} + 6q^{17} - 5q^{19} + O(q^{20})$$ ## Isogeny matrix

sage: E.isogeny_class().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 LMFDB numbering.

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

## Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)

The vertices are labelled with LMFDB labels. 