# Properties

 Label 3234.q Number of curves $4$ Conductor $3234$ CM no Rank $1$ Graph

# Related objects

Show commands: SageMath
E = EllipticCurve("q1")

E.isogeny_class()

## Elliptic curves in class 3234.q

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
3234.q1 3234p4 $$[1, 1, 1, -692518, 221512829]$$ $$312196988566716625/25367712678$$ $$2984486028854022$$ $$[2]$$ $$27648$$ $$2.0145$$
3234.q2 3234p3 $$[1, 1, 1, -40328, 3942245]$$ $$-61653281712625/21875235228$$ $$-2573599549338972$$ $$[2]$$ $$13824$$ $$1.6679$$
3234.q3 3234p2 $$[1, 1, 1, -17788, -465403]$$ $$5290763640625/2291573592$$ $$269601341525208$$ $$[2]$$ $$9216$$ $$1.4652$$
3234.q4 3234p1 $$[1, 1, 1, 3772, -51451]$$ $$50447927375/39517632$$ $$-4649209887168$$ $$[2]$$ $$4608$$ $$1.1186$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form3234.2.a.q

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.