# Properties

 Label 1764.e Number of curves $4$ Conductor $1764$ CM $$\Q(\sqrt{-3})$$ Rank $1$ Graph # Learn more

Show commands: SageMath
sage: E = EllipticCurve("e1")

sage: E.isogeny_class()

## Elliptic curves in class 1764.e

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
1764.e1 1764b4 $$[0, 0, 0, -6615, 203742]$$ $$54000$$ $$592815428352$$ $$$$ $$1728$$ $$1.0534$$   $$-12$$
1764.e2 1764b2 $$[0, 0, 0, -735, -7546]$$ $$54000$$ $$813189888$$ $$$$ $$576$$ $$0.50411$$   $$-12$$
1764.e3 1764b1 $$[0, 0, 0, 0, -343]$$ $$0$$ $$-50824368$$ $$$$ $$288$$ $$0.15754$$ $$\Gamma_0(N)$$-optimal $$-3$$
1764.e4 1764b3 $$[0, 0, 0, 0, 9261]$$ $$0$$ $$-37050964272$$ $$$$ $$864$$ $$0.70685$$   $$-3$$

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form1764.2.a.e

sage: E.q_eigenform(10)

$$q - 2q^{13} - 8q^{19} + O(q^{20})$$ ## 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.

$$\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 