# Properties

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

Show commands: SageMath
sage: E = EllipticCurve("h1")

sage: E.isogeny_class()

## Elliptic curves in class 2205.h

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
2205.h1 2205c2 $$[1, -1, 0, -73509, 5506388]$$ $$55306341/15625$$ $$12410625727828125$$ $$$$ $$16128$$ $$1.7950$$
2205.h2 2205c1 $$[1, -1, 0, -27204, -1652365]$$ $$2803221/125$$ $$99285005822625$$ $$$$ $$8064$$ $$1.4484$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form2205.2.a.h

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 LMFDB 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 LMFDB labels. 