# Properties

 Label 405.e Number of curves $2$ Conductor $405$ CM no Rank $1$ Graph

# Related objects

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

E.isogeny_class()

## Elliptic curves in class 405.e

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
405.e1 405c2 $$[1, -1, 0, -225, -1250]$$ $$-15590912409/78125$$ $$-6328125$$ $$[]$$ $$84$$ $$0.15237$$
405.e2 405c1 $$[1, -1, 0, 0, 1]$$ $$-9/5$$ $$-405$$ $$[]$$ $$12$$ $$-0.82059$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 405.e have rank $$1$$.

## Complex multiplication

The elliptic curves in class 405.e do not have complex multiplication.

## Modular form405.2.a.e

sage: E.q_eigenform(10)

$$q + q^{2} - q^{4} - q^{5} - 3 q^{7} - 3 q^{8} - q^{10} - 2 q^{11} - 2 q^{13} - 3 q^{14} - q^{16} + 4 q^{17} - 8 q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.