# Properties

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

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

E.isogeny_class()

## Elliptic curves in class 57.b

sage: E.isogeny_class().curves

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

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form57.2.a.b

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 