# Properties

 Label 10001.a Number of curves $2$ Conductor $10001$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("10001.a1")

sage: E.isogeny_class()

## Elliptic curves in class 10001.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
10001.a1 10001a2 [1, -1, 0, -53959, 4720704]  46656
10001.a2 10001a1 [1, -1, 0, -53594, 4788959]  23328 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 10001.a have rank $$1$$.

## Modular form 10001.2.a.a

sage: E.q_eigenform(10)

$$q + q^{2} - q^{4} - 4q^{5} - 4q^{7} - 3q^{8} - 3q^{9} - 4q^{10} + 4q^{11} - 4q^{14} - q^{16} + 2q^{17} - 3q^{18} - 4q^{19} + O(q^{20})$$

## 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 LMFDB numbering.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 