# Properties

 Label 10010f Number of curves 4 Conductor 10010 CM no Rank 0 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("10010.a1")

sage: E.isogeny_class()

## Elliptic curves in class 10010f

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
10010.a4 10010f1 [1, 0, 1, -2413624, 1442906886]  241920 $$\Gamma_0(N)$$-optimal
10010.a3 10010f2 [1, 0, 1, -2633144, 1164731142]  483840
10010.a2 10010f3 [1, 0, 1, -5585639, -3040342038]  725760
10010.a1 10010f4 [1, 0, 1, -78985959, -270129426454]  1451520

## Rank

sage: E.rank()

The elliptic curves in class 10010f have rank $$0$$.

## Modular form 10010.2.a.a

sage: E.q_eigenform(10)

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

## 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. 