# Properties

 Label 1728.n Number of curves $4$ Conductor $1728$ CM $$\Q(\sqrt{-3})$$ Rank $1$ Graph

# Related objects

Show commands: SageMath
sage: E = EllipticCurve("n1")

sage: E.isogeny_class()

## Elliptic curves in class 1728.n

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
1728.n1 1728a4 $$[0, 0, 0, -1080, -13662]$$ $$-12288000$$ $$-11337408$$ $$[]$$ $$432$$ $$0.39872$$   $$-27$$
1728.n2 1728a3 $$[0, 0, 0, -120, 506]$$ $$-12288000$$ $$-15552$$ $$[]$$ $$144$$ $$-0.15058$$   $$-27$$
1728.n3 1728a2 $$[0, 0, 0, 0, -54]$$ $$0$$ $$-1259712$$ $$[]$$ $$144$$ $$-0.15058$$   $$-3$$
1728.n4 1728a1 $$[0, 0, 0, 0, 2]$$ $$0$$ $$-1728$$ $$[]$$ $$48$$ $$-0.69989$$ $$\Gamma_0(N)$$-optimal $$-3$$

## Rank

sage: E.rank()

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

## Complex multiplication

Each elliptic curve in class 1728.n has complex multiplication by an order in the imaginary quadratic field $$\Q(\sqrt{-3})$$.

## Modular form1728.2.a.n

sage: E.q_eigenform(10)

$$q - q^{7} - 5q^{13} + 7q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrr} 1 & 27 & 3 & 9 \\ 27 & 1 & 9 & 3 \\ 3 & 9 & 1 & 3 \\ 9 & 3 & 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.