# Properties

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

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

sage: E.isogeny_class()

## Elliptic curves in class 20a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
20.a4 20a1 $$[0, 1, 0, 4, 4]$$ $$21296/25$$ $$-6400$$ $$$$ $$1$$ $$-0.58338$$ $$\Gamma_0(N)$$-optimal
20.a3 20a2 $$[0, 1, 0, -1, 0]$$ $$16384/5$$ $$80$$ $$$$ $$2$$ $$-0.92995$$
20.a2 20a3 $$[0, 1, 0, -36, -140]$$ $$-20720464/15625$$ $$-4000000$$ $$$$ $$3$$ $$-0.034070$$
20.a1 20a4 $$[0, 1, 0, -41, -116]$$ $$488095744/125$$ $$2000$$ $$$$ $$6$$ $$-0.38064$$

## Rank

sage: E.rank()

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

## Complex multiplication

The elliptic curves in class 20a do not have complex multiplication.

## 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 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. 