# Properties

 Label 57c Number of curves $2$ Conductor $57$ CM no Rank $0$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("c1")

sage: E.isogeny_class()

## Elliptic curves in class 57c

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
57.b2 57c1 $$[0, 1, 1, 20, -32]$$ $$841232384/1121931$$ $$-1121931$$ $$$$ $$12$$ $$-0.15331$$ $$\Gamma_0(N)$$-optimal
57.b1 57c2 $$[0, 1, 1, -4390, -113432]$$ $$-9358714467168256/22284891$$ $$-22284891$$ $$[]$$ $$60$$ $$0.65141$$

## Rank

sage: E.rank()

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

## Complex multiplication

The elliptic curves in class 57c do not have complex multiplication.

## Modular form57.2.a.c

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 Cremona 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 Cremona labels. 