# Properties

 Label 570.i Number of curves $4$ Conductor $570$ CM no Rank $0$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("570.i1")

sage: E.isogeny_class()

## Elliptic curves in class 570.i

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
570.i1 570i4 [1, 1, 1, -492480, 132819117]  3840
570.i2 570i3 [1, 1, 1, -31160, 2011565]  3840
570.i3 570i2 [1, 1, 1, -30780, 2065677] [2, 2] 1920
570.i4 570i1 [1, 1, 1, -1900, 32525]  960 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

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

## Modular form570.2.a.i

sage: E.q_eigenform(10)

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

## 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 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. 