# Properties

 Label 20.a Number of curves 4 Conductor 20 CM no Rank 0 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("20.a1")

sage: E.isogeny_class()

## Elliptic curves in class 20.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
20.a1 20a4 [0, 1, 0, -41, -116]  6
20.a2 20a3 [0, 1, 0, -36, -140]  3
20.a3 20a2 [0, 1, 0, -1, 0]  2
20.a4 20a1 [0, 1, 0, 4, 4]  1 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 20.a have rank $$0$$.

## Modular form20.2.a.a

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 LMFDB 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 LMFDB labels. 