# Properties

 Label 443760.df Number of curves $4$ Conductor $443760$ CM no Rank $1$ Graph # Learn more about

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

sage: E.isogeny_class()

## Elliptic curves in class 443760.df

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
443760.df1 443760df3 $$[0, 1, 0, -24806800, 47223347348]$$ $$65202655558249/512820150$$ $$13278094733281842585600$$ $$$$ $$34062336$$ $$3.0732$$ $$\Gamma_0(N)$$-optimal*
443760.df2 443760df2 $$[0, 1, 0, -2618800, -409851052]$$ $$76711450249/41602500$$ $$1077184537583708160000$$ $$[2, 2]$$ $$17031168$$ $$2.7267$$ $$\Gamma_0(N)$$-optimal*
443760.df3 443760df1 $$[0, 1, 0, -2027120, -1110163500]$$ $$35578826569/51600$$ $$1336042837313126400$$ $$$$ $$8515584$$ $$2.3801$$ $$\Gamma_0(N)$$-optimal*
443760.df4 443760df4 $$[0, 1, 0, 10102320, -3213585900]$$ $$4403686064471/2721093750$$ $$-70455383998934400000000$$ $$$$ $$34062336$$ $$3.0732$$
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 3 curves highlighted, and conditionally curve 443760.df1.

## Rank

sage: E.rank()

The elliptic curves in class 443760.df have rank $$1$$.

## Complex multiplication

The elliptic curves in class 443760.df do not have complex multiplication.

## Modular form 443760.2.a.df

sage: E.q_eigenform(10)

$$q + q^{3} + q^{5} + 4q^{7} + q^{9} - 2q^{13} + q^{15} + 2q^{17} + 4q^{19} + O(q^{20})$$ ## 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 LMFDB 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 LMFDB labels. 