# Properties

 Label 115920df Number of curves $4$ Conductor $115920$ CM no Rank $0$ Graph

# Related objects

Show commands: SageMath
sage: E = EllipticCurve("df1")

sage: E.isogeny_class()

## Elliptic curves in class 115920df

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
115920.e3 115920df1 $$[0, 0, 0, -31709163, 68726631962]$$ $$1180838681727016392361/692428800000$$ $$2067581317939200000$$ $$[2]$$ $$5898240$$ $$2.8385$$ $$\Gamma_0(N)$$-optimal
115920.e2 115920df2 $$[0, 0, 0, -31893483, 67887201818]$$ $$1201550658189465626281/28577902500000000$$ $$85333159618560000000000$$ $$[2, 2]$$ $$11796480$$ $$3.1851$$
115920.e4 115920df3 $$[0, 0, 0, 4106517, 212484801818]$$ $$2564821295690373719/6533572090396050000$$ $$-19509141724769158963200000$$ $$[2]$$ $$23592960$$ $$3.5317$$
115920.e1 115920df4 $$[0, 0, 0, -70842603, -130433927398]$$ $$13167998447866683762601/5158996582031250000$$ $$15404681250000000000000000$$ $$[2]$$ $$23592960$$ $$3.5317$$

## Rank

sage: E.rank()

The elliptic curves in class 115920df have rank $$0$$.

## Complex multiplication

The elliptic curves in class 115920df do not have complex multiplication.

## Modular form 115920.2.a.df

sage: E.q_eigenform(10)

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

## 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.