# Properties

 Label 1764.d Number of curves $2$ Conductor $1764$ CM $$\Q(\sqrt{-3})$$ Rank $1$ Graph

# Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 1764.d

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
1764.d1 1764c1 $$[0, 0, 0, 0, -28]$$ $$0$$ $$-338688$$ $$[]$$ $$144$$ $$-0.26005$$ $$\Gamma_0(N)$$-optimal $$-3$$
1764.d2 1764c2 $$[0, 0, 0, 0, 756]$$ $$0$$ $$-246903552$$ $$[]$$ $$432$$ $$0.28926$$   $$-3$$

## Rank

sage: E.rank()

The elliptic curves in class 1764.d have rank $$1$$.

## Complex multiplication

Each elliptic curve in class 1764.d has complex multiplication by an order in the imaginary quadratic field $$\Q(\sqrt{-3})$$.

## Modular form1764.2.a.d

sage: E.q_eigenform(10)

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

## 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.