# Properties

 Label 510.f Number of curves $4$ Conductor $510$ CM no Rank $0$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("510.f1")

sage: E.isogeny_class()

## Elliptic curves in class 510.f

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
510.f1 510f3 [1, 0, 0, -186, -990]  128
510.f2 510f4 [1, 0, 0, -166, 806]  128
510.f3 510f2 [1, 0, 0, -16, -4] [2, 2] 64
510.f4 510f1 [1, 0, 0, 4, 0]  32 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 510.f have rank $$0$$.

## Modular form510.2.a.f

sage: E.q_eigenform(10)

$$q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + q^{8} + q^{9} - q^{10} + 4q^{11} + q^{12} + 2q^{13} - q^{15} + q^{16} + q^{17} + q^{18} - 4q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 