# Properties

 Label 162240de Number of curves $4$ Conductor $162240$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("de1")

sage: E.isogeny_class()

## Elliptic curves in class 162240de

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
162240.ea4 162240de1 $$[0, -1, 0, -13745, -2196975]$$ $$-3631696/24375$$ $$-1927634442240000$$ $$$$ $$1032192$$ $$1.6163$$ $$\Gamma_0(N)$$-optimal
162240.ea3 162240de2 $$[0, -1, 0, -351745, -80004575]$$ $$15214885924/38025$$ $$12028438919577600$$ $$[2, 2]$$ $$2064384$$ $$1.9629$$
162240.ea2 162240de3 $$[0, -1, 0, -486945, -12702015]$$ $$20183398562/11567205$$ $$7318102238671011840$$ $$$$ $$4128768$$ $$2.3094$$
162240.ea1 162240de4 $$[0, -1, 0, -5624545, -5132401535]$$ $$31103978031362/195$$ $$123368604303360$$ $$$$ $$4128768$$ $$2.3094$$

## Rank

sage: E.rank()

The elliptic curves in class 162240de have rank $$1$$.

## Complex multiplication

The elliptic curves in class 162240de do not have complex multiplication.

## Modular form 162240.2.a.de

sage: E.q_eigenform(10)

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