# Properties

 Label 57.c Number of curves 4 Conductor 57 CM no Rank 0 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("57.c1")
sage: E.isogeny_class()

## Elliptic curves in class 57.c

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
57.c1 57b3 [1, 0, 1, -102, 385] 4 6
57.c2 57b1 [1, 0, 1, -7, 5] 4 3 $$\Gamma_0(N)$$-optimal
57.c3 57b2 [1, 0, 1, -2, -1] 2 6
57.c4 57b4 [1, 0, 1, 8, 29] 2 6

## Rank

sage: E.rank()

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

## Modular form57.2.a.c

sage: E.q_eigenform(10)
$$q + q^{2} + q^{3} - q^{4} - 2q^{5} + q^{6} - 3q^{8} + q^{9} - 2q^{10} - q^{12} + 6q^{13} - 2q^{15} - q^{16} - 6q^{17} + q^{18} - q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.