# Properties

 Label 80a Number of curves 4 Conductor 80 CM no Rank 0 Graph

# Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("80.a1")

sage: E.isogeny_class()

## Elliptic curves in class 80a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
80.a2 80a1 [0, 0, 0, -7, 6] [2, 2] 4 $$\Gamma_0(N)$$-optimal
80.a3 80a2 [0, 0, 0, -2, -1] [2] 8
80.a1 80a3 [0, 0, 0, -107, 426] [4] 8
80.a4 80a4 [0, 0, 0, 13, 34] [4] 8

## Rank

sage: E.rank()

The elliptic curves in class 80a have rank $$0$$.

## Modular form80.2.a.a

sage: E.q_eigenform(10)

$$q + q^{5} + 4q^{7} - 3q^{9} - 4q^{11} - 2q^{13} + 2q^{17} - 4q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.