# Properties

 Label 510.e Number of curves $8$ Conductor $510$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("510.e1")

sage: E.isogeny_class()

## Elliptic curves in class 510.e

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
510.e1 510e7 [1, 1, 1, -113470, -14759143] [2] 2048
510.e2 510e4 [1, 1, 1, -21760, 1226417] [4] 512
510.e3 510e5 [1, 1, 1, -7220, -224143] [2, 2] 1024
510.e4 510e3 [1, 1, 1, -1440, 16305] [2, 4] 512
510.e5 510e2 [1, 1, 1, -1360, 18737] [2, 4] 256
510.e6 510e1 [1, 1, 1, -80, 305] [4] 128 $$\Gamma_0(N)$$-optimal
510.e7 510e6 [1, 1, 1, 3060, 102705] [4] 1024
510.e8 510e8 [1, 1, 1, 6550, -962215] [2] 2048

## Rank

sage: E.rank()

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

## Modular form510.2.a.e

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.

$$\left(\begin{array}{rrrrrrrr} 1 & 16 & 2 & 4 & 8 & 16 & 8 & 4 \\ 16 & 1 & 8 & 4 & 2 & 4 & 8 & 16 \\ 2 & 8 & 1 & 2 & 4 & 8 & 4 & 2 \\ 4 & 4 & 2 & 1 & 2 & 4 & 2 & 4 \\ 8 & 2 & 4 & 2 & 1 & 2 & 4 & 8 \\ 16 & 4 & 8 & 4 & 2 & 1 & 8 & 16 \\ 8 & 8 & 4 & 2 & 4 & 8 & 1 & 8 \\ 4 & 16 & 2 & 4 & 8 & 16 & 8 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.