# Properties

 Label 2205d 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 2205d

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
2205.c2 2205d1 $$[1, -1, 1, -62, -164]$$ $$2803221/125$$ $$1157625$$ $$$$ $$384$$ $$-0.073855$$ $$\Gamma_0(N)$$-optimal
2205.c1 2205d2 $$[1, -1, 1, -167, 634]$$ $$55306341/15625$$ $$144703125$$ $$$$ $$768$$ $$0.27272$$

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form2205.2.a.d

sage: E.q_eigenform(10)

$$q - q^{2} - q^{4} + q^{5} + 3q^{8} - q^{10} + 2q^{11} - 6q^{13} - q^{16} - 6q^{17} + 6q^{19} + O(q^{20})$$ ## 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 Cremona numbering.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels. 