# Properties

 Label 6720y Number of curves $4$ Conductor $6720$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("y1")

sage: E.isogeny_class()

## Elliptic curves in class 6720y

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
6720.cg3 6720y1 [0, 1, 0, -225, -897]  3072 $$\Gamma_0(N)$$-optimal
6720.cg2 6720y2 [0, 1, 0, -1505, 21375] [2, 2] 6144
6720.cg1 6720y3 [0, 1, 0, -23905, 1414655]  12288
6720.cg4 6720y4 [0, 1, 0, 415, 73983]  12288

## Rank

sage: E.rank()

The elliptic curves in class 6720y have rank $$1$$.

## Complex multiplication

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

## Modular form6720.2.a.y

sage: E.q_eigenform(10)

$$q + q^{3} + q^{5} - q^{7} + q^{9} + 4q^{11} + 2q^{13} + q^{15} - 6q^{17} + O(q^{20})$$

## 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 Cremona 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 Cremona labels. 