# Properties

 Label 335730v Number of curves $6$ Conductor $335730$ CM no Rank $2$ Graph

# Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("335730.v1")

sage: E.isogeny_class()

## Elliptic curves in class 335730v

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
335730.v6 335730v1 [1, 1, 0, 21653, 6957181] [2] 3538944 $$\Gamma_0(N)$$-optimal
335730.v5 335730v2 [1, 1, 0, -440427, 106119549] [2, 2] 7077888
335730.v2 335730v3 [1, 1, 0, -6938427, 7031687949] [2, 2] 14155776
335730.v4 335730v4 [1, 1, 0, -1335707, -463815699] [2] 14155776
335730.v1 335730v5 [1, 1, 0, -111014727, 450167758089] [2] 28311552
335730.v3 335730v6 [1, 1, 0, -6830127, 7261955409] [2] 28311552

## Rank

sage: E.rank()

The elliptic curves in class 335730v have rank $$2$$.

## Modular form 335730.2.a.v

sage: E.q_eigenform(10)

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

## 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 Cremona numbering.

$$\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels.