# Properties

 Label 720.c Number of curves $8$ Conductor $720$ CM no Rank $1$ Graph

# Related objects

Show commands: SageMath
E = EllipticCurve("c1")

E.isogeny_class()

## Elliptic curves in class 720.c

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
720.c1 720h7 $$[0, 0, 0, -311043, -66769598]$$ $$1114544804970241/405$$ $$1209323520$$ $$[2]$$ $$2048$$ $$1.5333$$
720.c2 720h5 $$[0, 0, 0, -19443, -1042958]$$ $$272223782641/164025$$ $$489776025600$$ $$[2, 2]$$ $$1024$$ $$1.1867$$
720.c3 720h8 $$[0, 0, 0, -15843, -1441118]$$ $$-147281603041/215233605$$ $$-642684100792320$$ $$[2]$$ $$2048$$ $$1.5333$$
720.c4 720h4 $$[0, 0, 0, -11523, 476098]$$ $$56667352321/15$$ $$44789760$$ $$[2]$$ $$512$$ $$0.84018$$
720.c5 720h3 $$[0, 0, 0, -1443, -9758]$$ $$111284641/50625$$ $$151165440000$$ $$[2, 2]$$ $$512$$ $$0.84018$$
720.c6 720h2 $$[0, 0, 0, -723, 7378]$$ $$13997521/225$$ $$671846400$$ $$[2, 2]$$ $$256$$ $$0.49360$$
720.c7 720h1 $$[0, 0, 0, -3, 322]$$ $$-1/15$$ $$-44789760$$ $$[2]$$ $$128$$ $$0.14703$$ $$\Gamma_0(N)$$-optimal
720.c8 720h6 $$[0, 0, 0, 5037, -73262]$$ $$4733169839/3515625$$ $$-10497600000000$$ $$[2]$$ $$1024$$ $$1.1867$$

## Rank

sage: E.rank()

The elliptic curves in class 720.c have rank $$1$$.

## Complex multiplication

The elliptic curves in class 720.c do not have complex multiplication.

## Modular form720.2.a.c

sage: E.q_eigenform(10)

$$q - q^{5} - 4 q^{11} - 2 q^{13} - 2 q^{17} - 4 q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 16 & 4 & 8 & 16 & 8 \\ 2 & 1 & 2 & 8 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 16 & 4 & 8 & 16 & 8 \\ 16 & 8 & 16 & 1 & 4 & 2 & 4 & 8 \\ 4 & 2 & 4 & 4 & 1 & 2 & 4 & 2 \\ 8 & 4 & 8 & 2 & 2 & 1 & 2 & 4 \\ 16 & 8 & 16 & 4 & 4 & 2 & 1 & 8 \\ 8 & 4 & 8 & 8 & 2 & 4 & 8 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.