# Properties

 Label 10005d Number of curves 4 Conductor 10005 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("10005.k1")

sage: E.isogeny_class()

## Elliptic curves in class 10005d

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
10005.k4 10005d1 [1, 1, 0, 43, 216]  1920 $$\Gamma_0(N)$$-optimal
10005.k3 10005d2 [1, 1, 0, -362, 2079] [2, 2] 3840
10005.k2 10005d3 [1, 1, 0, -1667, -24804]  7680
10005.k1 10005d4 [1, 1, 0, -5537, 156294]  7680

## Rank

sage: E.rank()

The elliptic curves in class 10005d have rank $$1$$.

## Modular form 10005.2.a.k

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels. 