# Properties

 Label 405.d Number of curves $2$ Conductor $405$ CM no Rank $0$ Graph

# Related objects

Show commands: SageMath
E = EllipticCurve("d1")

E.isogeny_class()

## Elliptic curves in class 405.d

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
405.d1 405a2 $$[0, 0, 1, -162, -790]$$ $$884736/5$$ $$2657205$$ $$[]$$ $$72$$ $$0.074670$$
405.d2 405a1 $$[0, 0, 1, -12, 15]$$ $$2359296/125$$ $$10125$$ $$[3]$$ $$24$$ $$-0.47464$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 405.d have rank $$0$$.

## Complex multiplication

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

## Modular form405.2.a.d

sage: E.q_eigenform(10)

$$q - 2 q^{4} + q^{5} + 2 q^{7} + 3 q^{11} - 4 q^{13} + 4 q^{16} + 6 q^{17} - q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.