# Properties

 Label 570.e Number of curves $4$ Conductor $570$ CM no Rank $1$ Graph # Learn more

Show commands for: SageMath
sage: E = EllipticCurve("e1")

sage: E.isogeny_class()

## Elliptic curves in class 570.e

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
570.e1 570e3 $$[1, 0, 1, -968, -11662]$$ $$100162392144121/23457780$$ $$23457780$$ $$$$ $$512$$ $$0.40469$$
570.e2 570e4 $$[1, 0, 1, -448, 3506]$$ $$9912050027641/311647500$$ $$311647500$$ $$$$ $$512$$ $$0.40469$$
570.e3 570e2 $$[1, 0, 1, -68, -142]$$ $$34043726521/11696400$$ $$11696400$$ $$[2, 2]$$ $$256$$ $$0.058114$$
570.e4 570e1 $$[1, 0, 1, 12, -14]$$ $$214921799/218880$$ $$-218880$$ $$$$ $$128$$ $$-0.28846$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 570.e have rank $$1$$.

## Complex multiplication

The elliptic curves in class 570.e do not have complex multiplication.

## Modular form570.2.a.e

sage: E.q_eigenform(10)

$$q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} - 4q^{7} - q^{8} + q^{9} - q^{10} - 4q^{11} + q^{12} - 6q^{13} + 4q^{14} + q^{15} + q^{16} - 6q^{17} - q^{18} - q^{19} + O(q^{20})$$ ## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 