# Properties

 Label 20.a Number of curves $4$ Conductor $20$ CM no Rank $0$ Graph

# Related objects

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

E.isogeny_class()

## Elliptic curves in class 20.a

sage: E.isogeny_class().curves

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

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form20.2.a.a

sage: E.q_eigenform(10)

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

## 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.