# Properties

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

Show commands: SageMath
E = EllipticCurve("b1")

E.isogeny_class()

## Elliptic curves in class 58.b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
58.b1 58b2 $$[1, 1, 1, -455, -3951]$$ $$-10418796526321/82044596$$ $$-82044596$$ $$[]$$ $$20$$ $$0.34716$$
58.b2 58b1 $$[1, 1, 1, 5, 9]$$ $$13651919/29696$$ $$-29696$$ $$$$ $$4$$ $$-0.45755$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

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

## Complex multiplication

The elliptic curves in class 58.b do not have complex multiplication.

## Modular form58.2.a.b

sage: E.q_eigenform(10)

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

## 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. 