# Properties

 Label 115920cy Number of curves $4$ Conductor $115920$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("cy1")

sage: E.isogeny_class()

## Elliptic curves in class 115920cy

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
115920.a3 115920cy1 $$[0, 0, 0, -69123, 398200322]$$ $$-12232183057921/22933241856000$$ $$-68478293250146304000$$ $$$$ $$3981312$$ $$2.4851$$ $$\Gamma_0(N)$$-optimal
115920.a2 115920cy2 $$[0, 0, 0, -8547843, 9502649858]$$ $$23131609187144855041/322060536000000$$ $$961667607527424000000$$ $$$$ $$7962624$$ $$2.8316$$
115920.a4 115920cy3 $$[0, 0, 0, 622077, -10748505598]$$ $$8915971454369279/16719623332762560$$ $$-49924527757655679959040$$ $$$$ $$11943936$$ $$3.0344$$
115920.a1 115920cy4 $$[0, 0, 0, -69459843, -218064841342]$$ $$12411881707829361287041/303132494474220600$$ $$905148778380111124070400$$ $$$$ $$23887872$$ $$3.3809$$

## Rank

sage: E.rank()

The elliptic curves in class 115920cy have rank $$1$$.

## Complex multiplication

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

## Modular form 115920.2.a.cy

sage: E.q_eigenform(10)

$$q - q^{5} - q^{7} - 6q^{11} - 4q^{13} + 6q^{17} + 4q^{19} + O(q^{20})$$ ## 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 Cremona numbering.

$$\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels. 