# Properties

 Label 10580m Number of curves 4 Conductor 10580 CM no Rank 0 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("10580.c1")

sage: E.isogeny_class()

## Elliptic curves in class 10580m

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
10580.c3 10580m1 [0, 1, 0, -705, -5192]  5940 $$\Gamma_0(N)$$-optimal
10580.c4 10580m2 [0, 1, 0, 1940, -32700]  11880
10580.c1 10580m3 [0, 1, 0, -21865, 1236900]  17820
10580.c2 10580m4 [0, 1, 0, -19220, 1550068]  35640

## Rank

sage: E.rank()

The elliptic curves in class 10580m have rank $$0$$.

## Modular form 10580.2.a.c

sage: E.q_eigenform(10)

$$q - 2q^{3} + q^{5} - 2q^{7} + q^{9} + 2q^{13} - 2q^{15} + 6q^{17} + 4q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels. 