# Properties

 Label 75712.cn Number of curves $2$ Conductor $75712$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("cn1")

sage: E.isogeny_class()

## Elliptic curves in class 75712.cn

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
75712.cn1 75712cx2 $$[0, -1, 0, -27265, 1721121]$$ $$3543122/49$$ $$31000315953152$$ $$$$ $$301056$$ $$1.3944$$
75712.cn2 75712cx1 $$[0, -1, 0, -225, 71681]$$ $$-4/7$$ $$-2214308282368$$ $$$$ $$150528$$ $$1.0478$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 75712.cn have rank $$1$$.

## Complex multiplication

The elliptic curves in class 75712.cn do not have complex multiplication.

## Modular form 75712.2.a.cn

sage: E.q_eigenform(10)

$$q + 2q^{3} - 4q^{5} + q^{7} + q^{9} - 8q^{15} - 2q^{17} + 2q^{19} + O(q^{20})$$ ## 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.

$$\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 