# Properties

 Label 510.c Number of curves $4$ Conductor $510$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("510.c1")

sage: E.isogeny_class()

## Elliptic curves in class 510.c

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
510.c1 510d3 [1, 1, 1, -6541, -206341]  768
510.c2 510d2 [1, 1, 1, -421, -3157] [2, 2] 384
510.c3 510d1 [1, 1, 1, -101, 299]  192 $$\Gamma_0(N)$$-optimal
510.c4 510d4 [1, 1, 1, 579, -14757]  768

## Rank

sage: E.rank()

The elliptic curves in class 510.c have rank $$1$$.

## Modular form510.2.a.c

sage: E.q_eigenform(10)

$$q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} - 4q^{7} + q^{8} + q^{9} - q^{10} - 4q^{11} - q^{12} - 2q^{13} - 4q^{14} + q^{15} + q^{16} + q^{17} + q^{18} - 4q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 