# Properties

 Label 57b Number of curves $4$ Conductor $57$ CM no Rank $0$ Graph

# Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 57b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
57.c2 57b1 $$[1, 0, 1, -7, 5]$$ $$30664297/3249$$ $$3249$$ $$[2, 2]$$ $$3$$ $$-0.59158$$ $$\Gamma_0(N)$$-optimal
57.c3 57b2 $$[1, 0, 1, -2, -1]$$ $$389017/57$$ $$57$$ $$[2]$$ $$6$$ $$-0.93816$$
57.c1 57b3 $$[1, 0, 1, -102, 385]$$ $$115714886617/1539$$ $$1539$$ $$[4]$$ $$6$$ $$-0.24501$$
57.c4 57b4 $$[1, 0, 1, 8, 29]$$ $$67419143/390963$$ $$-390963$$ $$[2]$$ $$6$$ $$-0.24501$$

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form57.2.a.b

sage: E.q_eigenform(10)

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

## 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 Cremona numbering.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.