# Properties

 Label 714.e Number of curves $4$ Conductor $714$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("e1")

sage: E.isogeny_class()

## Elliptic curves in class 714.e

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
714.e1 714f4 $$[1, 1, 1, -5079, 137205]$$ $$14489843500598257/6246072$$ $$6246072$$ $$$$ $$768$$ $$0.64655$$
714.e2 714f3 $$[1, 1, 1, -679, -3883]$$ $$34623662831857/14438442312$$ $$14438442312$$ $$$$ $$768$$ $$0.64655$$
714.e3 714f2 $$[1, 1, 1, -319, 2021]$$ $$3590714269297/73410624$$ $$73410624$$ $$[2, 2]$$ $$384$$ $$0.29998$$
714.e4 714f1 $$[1, 1, 1, 1, 101]$$ $$103823/4386816$$ $$-4386816$$ $$$$ $$192$$ $$-0.046597$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 714.e have rank $$1$$.

## Complex multiplication

The elliptic curves in class 714.e do not have complex multiplication.

## Modular form714.2.a.e

sage: E.q_eigenform(10)

$$q + q^{2} - q^{3} + q^{4} - 2q^{5} - q^{6} - q^{7} + q^{8} + q^{9} - 2q^{10} - q^{12} - 6q^{13} - q^{14} + 2q^{15} + q^{16} + q^{17} + q^{18} + O(q^{20})$$ ## 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. 