# Properties

 Label 432.e Number of curves $4$ Conductor $432$ CM $$\Q(\sqrt{-3})$$ Rank $0$ Graph

# Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 432.e

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
432.e1 432a4 $$[0, 0, 0, -4320, 109296]$$ $$-12288000$$ $$-725594112$$ $$[]$$ $$216$$ $$0.74530$$   $$-27$$
432.e2 432a2 $$[0, 0, 0, -480, -4048]$$ $$-12288000$$ $$-995328$$ $$[]$$ $$72$$ $$0.19599$$   $$-27$$
432.e3 432a1 $$[0, 0, 0, 0, -16]$$ $$0$$ $$-110592$$ $$[]$$ $$24$$ $$-0.35332$$ $$\Gamma_0(N)$$-optimal $$-3$$
432.e4 432a3 $$[0, 0, 0, 0, 432]$$ $$0$$ $$-80621568$$ $$[]$$ $$72$$ $$0.19599$$   $$-3$$

## Rank

sage: E.rank()

The elliptic curves in class 432.e have rank $$0$$.

## Complex multiplication

Each elliptic curve in class 432.e has complex multiplication by an order in the imaginary quadratic field $$\Q(\sqrt{-3})$$.

## Modular form432.2.a.e

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.