# Properties

 Label 58.b Number of curves 2 Conductor 58 CM no Rank 0 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("58.b1")
sage: E.isogeny_class()

## Elliptic curves in class 58.b

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
58.b1 58b2 [1, 1, 1, -455, -3951] 1 20
58.b2 58b1 [1, 1, 1, 5, 9] 5 4 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 58.b have rank $$0$$.

## Modular form58.2.a.b

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

## 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.

$$\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 