# Properties

 Label 3234.p Number of curves $4$ Conductor $3234$ CM no Rank $0$ Graph # Related objects

Show commands: SageMath
E = EllipticCurve("p1")

E.isogeny_class()

## Elliptic curves in class 3234.p

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
3234.p1 3234r3 $$[1, 1, 1, -1972104, -1066786449]$$ $$7209828390823479793/49509306$$ $$5824720341594$$ $$$$ $$36864$$ $$2.0492$$
3234.p2 3234r4 $$[1, 1, 1, -171844, -2403745]$$ $$4770223741048753/2740574865798$$ $$322425892386268902$$ $$$$ $$36864$$ $$2.0492$$
3234.p3 3234r2 $$[1, 1, 1, -123334, -16685089]$$ $$1763535241378513/4612311396$$ $$542633823428004$$ $$[2, 2]$$ $$18432$$ $$1.7026$$
3234.p4 3234r1 $$[1, 1, 1, -4754, -463345]$$ $$-100999381393/723148272$$ $$-85077671052528$$ $$$$ $$9216$$ $$1.3560$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form3234.2.a.p

sage: E.q_eigenform(10)

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

## 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.

$$\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 