# Properties

 Label 6720.ca Number of curves $4$ Conductor $6720$ CM no Rank $0$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("ca1")

sage: E.isogeny_class()

## Elliptic curves in class 6720.ca

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
6720.ca1 6720ci3 $$[0, 1, 0, -89185, -6377665]$$ $$2394165105226952/854262178245$$ $$27992463056732160$$ $$$$ $$61440$$ $$1.8568$$
6720.ca2 6720ci2 $$[0, 1, 0, -79385, -8633625]$$ $$13507798771700416/3544416225$$ $$14517928857600$$ $$[2, 2]$$ $$30720$$ $$1.5103$$
6720.ca3 6720ci1 $$[0, 1, 0, -79380, -8634762]$$ $$864335783029582144/59535$$ $$3810240$$ $$$$ $$15360$$ $$1.1637$$ $$\Gamma_0(N)$$-optimal
6720.ca4 6720ci4 $$[0, 1, 0, -69665, -10816737]$$ $$-1141100604753992/875529151875$$ $$-28689339248640000$$ $$$$ $$61440$$ $$1.8568$$

## Rank

sage: E.rank()

The elliptic curves in class 6720.ca have rank $$0$$.

## Complex multiplication

The elliptic curves in class 6720.ca do not have complex multiplication.

## Modular form6720.2.a.ca

sage: E.q_eigenform(10)

$$q + q^{3} + q^{5} - q^{7} + q^{9} - 4 q^{11} + 6 q^{13} + q^{15} + 6 q^{17} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 