# Properties

 Label 1728.d Number of curves $3$ Conductor $1728$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("d1")

sage: E.isogeny_class()

## Elliptic curves in class 1728.d

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1728.d1 1728bb3 $$[0, 0, 0, -7884, 357264]$$ $$-1167051/512$$ $$-23776267862016$$ $$[]$$ $$3456$$ $$1.2738$$
1728.d2 1728bb1 $$[0, 0, 0, -204, -1136]$$ $$-132651/2$$ $$-14155776$$ $$[]$$ $$384$$ $$0.17521$$ $$\Gamma_0(N)$$-optimal
1728.d3 1728bb2 $$[0, 0, 0, 756, -5616]$$ $$9261/8$$ $$-41278242816$$ $$[]$$ $$1152$$ $$0.72452$$

## Rank

sage: E.rank()

The elliptic curves in class 1728.d have rank $$1$$.

## Complex multiplication

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

## Modular form1728.2.a.d

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. 