# Properties

 Label 80b Number of curves 4 Conductor 80 CM no Rank 0 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("80.b1")

sage: E.isogeny_class()

## Elliptic curves in class 80b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
80.b4 80b1 [0, -1, 0, 4, -4]  4 $$\Gamma_0(N)$$-optimal
80.b3 80b2 [0, -1, 0, -1, 0]  8
80.b2 80b3 [0, -1, 0, -36, 140]  12
80.b1 80b4 [0, -1, 0, -41, 116]  24

## Rank

sage: E.rank()

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

## Modular form80.2.a.b

sage: E.q_eigenform(10)

$$q + 2q^{3} - q^{5} - 2q^{7} + q^{9} + 2q^{13} - 2q^{15} - 6q^{17} + 4q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels. 