# Properties

 Label 57.b Number of curves 2 Conductor 57 CM no Rank 0 Graph # Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("57.b1")
sage: E.isogeny_class()

## Elliptic curves in class 57.b

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
57.b1 57c2 [0, 1, 1, -4390, -113432] 1 60
57.b2 57c1 [0, 1, 1, 20, -32] 5 12 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 57.b have rank $$0$$.

## Modular form57.2.a.b

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 