# Properties

 Label 3234.r Number of curves $2$ Conductor $3234$ CM no Rank $0$ Graph

# Related objects

Show commands: SageMath
E = EllipticCurve("r1")

E.isogeny_class()

## Elliptic curves in class 3234.r

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
3234.r1 3234q2 $$[1, 1, 1, -2570492, -1587302179]$$ $$46546832455691959/748268928$$ $$30195350250823296$$ $$[2]$$ $$75264$$ $$2.2951$$
3234.r2 3234q1 $$[1, 1, 1, -155772, -26427171]$$ $$-10358806345399/1445216256$$ $$-58319688824635392$$ $$[2]$$ $$37632$$ $$1.9485$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 3234.r have rank $$0$$.

## Complex multiplication

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

## Modular form3234.2.a.r

sage: E.q_eigenform(10)

$$q + q^{2} - q^{3} + q^{4} + 2 q^{5} - q^{6} + q^{8} + q^{9} + 2 q^{10} + q^{11} - q^{12} + 4 q^{13} - 2 q^{15} + q^{16} + 6 q^{17} + q^{18} + 2 q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.