# Properties

 Label 8005.a Number of curves 2 Conductor 8005 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("8005.a1")

sage: E.isogeny_class()

## Elliptic curves in class 8005.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
8005.a1 8005b1 [1, 0, 0, -25, 0]  1248 $$\Gamma_0(N)$$-optimal
8005.a2 8005b2 [1, 0, 0, 100, 25]  2496

## Rank

sage: E.rank()

The elliptic curves in class 8005.a have rank $$1$$.

## Modular form8005.2.a.a

sage: E.q_eigenform(10)

$$q - q^{2} - 2q^{3} - q^{4} + q^{5} + 2q^{6} + 2q^{7} + 3q^{8} + q^{9} - q^{10} - 6q^{11} + 2q^{12} - 2q^{13} - 2q^{14} - 2q^{15} - q^{16} + 6q^{17} - q^{18} - 4q^{19} + O(q^{20})$$

## 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 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. 