# Properties

 Label 2205i Number of curves $2$ Conductor $2205$ CM no Rank $1$ Graph

# Related objects

Show commands: SageMath
sage: E = EllipticCurve("i1")

sage: E.isogeny_class()

## Elliptic curves in class 2205i

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
2205.f2 2205i1 $$[0, 0, 1, 2058, 72630]$$ $$229376/675$$ $$-2836714452075$$ $$[]$$ $$2688$$ $$1.0729$$ $$\Gamma_0(N)$$-optimal
2205.f1 2205i2 $$[0, 0, 1, -90552, 10509777]$$ $$-19539165184/46875$$ $$-196994059171875$$ $$[3]$$ $$8064$$ $$1.6222$$

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form2205.2.a.i

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 Cremona 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 Cremona labels.