# Properties

 Label 10008f Number of curves 2 Conductor 10008 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("10008.f1")

sage: E.isogeny_class()

## Elliptic curves in class 10008f

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
10008.f1 10008f1 [0, 0, 0, -11235, -458354]  10752 $$\Gamma_0(N)$$-optimal
10008.f2 10008f2 [0, 0, 0, -10875, -489098]  21504

## Rank

sage: E.rank()

The elliptic curves in class 10008f have rank $$1$$.

## Modular form 10008.2.a.f

sage: E.q_eigenform(10)

$$q + 4q^{7} - 4q^{11} + 6q^{13} - 2q^{17} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels. 