# Properties

 Label 1728.m Number of curves $2$ Conductor $1728$ CM $$\Q(\sqrt{-3})$$ Rank $0$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("m1")

sage: E.isogeny_class()

## Elliptic curves in class 1728.m

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
1728.m1 1728m1 $$[0, 0, 0, 0, -32]$$ $$0$$ $$-442368$$ $$[]$$ $$192$$ $$-0.23779$$ $$\Gamma_0(N)$$-optimal $$-3$$
1728.m2 1728m2 $$[0, 0, 0, 0, 864]$$ $$0$$ $$-322486272$$ $$[]$$ $$576$$ $$0.31151$$   $$-3$$

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form1728.2.a.m

sage: E.q_eigenform(10)

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

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 