# Properties

 Label 570g Number of curves $4$ Conductor $570$ CM no Rank $0$ Graph

# Related objects

Show commands: SageMath
sage: E = EllipticCurve("g1")

sage: E.isogeny_class()

## Elliptic curves in class 570g

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
570.g3 570g1 $$[1, 1, 1, -31, 53]$$ $$3301293169/22800$$ $$22800$$ $$[4]$$ $$64$$ $$-0.33023$$ $$\Gamma_0(N)$$-optimal
570.g2 570g2 $$[1, 1, 1, -51, -51]$$ $$14688124849/8122500$$ $$8122500$$ $$[2, 2]$$ $$128$$ $$0.016344$$
570.g1 570g3 $$[1, 1, 1, -621, -6207]$$ $$26487576322129/44531250$$ $$44531250$$ $$[2]$$ $$256$$ $$0.36292$$
570.g4 570g4 $$[1, 1, 1, 199, -151]$$ $$871257511151/527800050$$ $$-527800050$$ $$[2]$$ $$256$$ $$0.36292$$

## Rank

sage: E.rank()

The elliptic curves in class 570g have rank $$0$$.

## Complex multiplication

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

## Modular form570.2.a.g

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.