# Properties

 Label 588.e Number of curves $2$ Conductor $588$ CM no Rank $0$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("588.e1")

sage: E.isogeny_class()

## Elliptic curves in class 588.e

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
588.e1 588e2 [0, 1, 0, -2172, -36828]  672
588.e2 588e1 [0, 1, 0, -457, 2960]  336 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 588.e have rank $$0$$.

## Modular form588.2.a.e

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 