# Properties

 Label 1728.z Number of curves $3$ Conductor $1728$ CM no Rank $0$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("z1")

sage: E.isogeny_class()

## Elliptic curves in class 1728.z

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1728.z1 1728s3 $$[0, 0, 0, -1836, 30672]$$ $$-132651/2$$ $$-10319560704$$ $$[]$$ $$1152$$ $$0.72452$$
1728.z2 1728s2 $$[0, 0, 0, -876, -13232]$$ $$-1167051/512$$ $$-32614907904$$ $$[]$$ $$1152$$ $$0.72452$$
1728.z3 1728s1 $$[0, 0, 0, 84, 208]$$ $$9261/8$$ $$-56623104$$ $$[]$$ $$384$$ $$0.17521$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 1728.z have rank $$0$$.

## Complex multiplication

The elliptic curves in class 1728.z do not have complex multiplication.

## Modular form1728.2.a.z

sage: E.q_eigenform(10)

$$q + 3q^{5} + q^{7} + 3q^{11} + 4q^{13} + 2q^{19} + O(q^{20})$$ ## 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.

$$\left(\begin{array}{rrr} 1 & 9 & 3 \\ 9 & 1 & 3 \\ 3 & 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 