# Properties

 Label 3234.e Number of curves $2$ Conductor $3234$ CM no Rank $1$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 3234.e

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
3234.e1 3234g2 $$[1, 1, 0, -72265, 7447189]$$ $$121681065322255375/12702096$$ $$4356818928$$ $$$$ $$8192$$ $$1.2785$$
3234.e2 3234g1 $$[1, 1, 0, -4505, 115557]$$ $$-29489309167375/303595776$$ $$-104133351168$$ $$$$ $$4096$$ $$0.93194$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 3234.e have rank $$1$$.

## Complex multiplication

The elliptic curves in class 3234.e do not have complex multiplication.

## Modular form3234.2.a.e

sage: E.q_eigenform(10)

$$q - q^{2} - q^{3} + q^{4} + q^{6} - q^{8} + q^{9} + q^{11} - q^{12} - 4 q^{13} + q^{16} + 4 q^{17} - q^{18} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 