# Properties

 Label 2205.e Number of curves $3$ Conductor $2205$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("e1")

sage: E.isogeny_class()

## Elliptic curves in class 2205.e

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
2205.e1 2205g3 $$[0, 0, 1, -57918, -5612252]$$ $$-250523582464/13671875$$ $$-1172583685546875$$ $$[]$$ $$8640$$ $$1.6497$$
2205.e2 2205g1 $$[0, 0, 1, -588, 6088]$$ $$-262144/35$$ $$-3001814235$$ $$[]$$ $$960$$ $$0.55111$$ $$\Gamma_0(N)$$-optimal
2205.e3 2205g2 $$[0, 0, 1, 3822, -15521]$$ $$71991296/42875$$ $$-3677222437875$$ $$[]$$ $$2880$$ $$1.1004$$

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form2205.2.a.e

sage: E.q_eigenform(10)

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

$$\left(\begin{array}{rrr} 1 & 9 & 3 \\ 9 & 1 & 3 \\ 3 & 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 