# Properties

 Label 121680.fo Number of curves $4$ Conductor $121680$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("fo1")

sage: E.isogeny_class()

## Elliptic curves in class 121680.fo

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
121680.fo1 121680bv4 $$[0, 0, 0, -12655227, 17328182794]$$ $$31103978031362/195$$ $$1405245508392960$$ $$$$ $$4128768$$ $$2.5122$$
121680.fo2 121680bv3 $$[0, 0, 0, -1095627, 43417114]$$ $$20183398562/11567205$$ $$83357758312361994240$$ $$$$ $$4128768$$ $$2.5122$$
121680.fo3 121680bv2 $$[0, 0, 0, -791427, 270411154]$$ $$15214885924/38025$$ $$137011437068313600$$ $$[2, 2]$$ $$2064384$$ $$2.1656$$
121680.fo4 121680bv1 $$[0, 0, 0, -30927, 7430254]$$ $$-3631696/24375$$ $$-21956961068640000$$ $$$$ $$1032192$$ $$1.8190$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 121680.fo have rank $$1$$.

## Complex multiplication

The elliptic curves in class 121680.fo do not have complex multiplication.

## Modular form 121680.2.a.fo

sage: E.q_eigenform(10)

$$q + q^{5} + 4q^{7} - 4q^{11} - 6q^{17} + O(q^{20})$$ ## 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 LMFDB numbering.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 