# Properties

 Label 350658.dw Number of curves $6$ Conductor $350658$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("350658.dw1")

sage: E.isogeny_class()

## Elliptic curves in class 350658.dw

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
350658.dw1 350658dw6 [1, -1, 1, -86176949, -307893151695] [2] 41943040
350658.dw2 350658dw3 [1, -1, 1, -19290569, 32615063385] [2] 20971520
350658.dw3 350658dw4 [1, -1, 1, -5525609, -4547331687] [2, 2] 20971520
350658.dw4 350658dw2 [1, -1, 1, -1256729, 464333433] [2, 2] 10485760
350658.dw5 350658dw1 [1, -1, 1, 137191, 40024185] [2] 5242880 $$\Gamma_0(N)$$-optimal
350658.dw6 350658dw5 [1, -1, 1, 6823651, -21999305919] [2] 41943040

## Rank

sage: E.rank()

The elliptic curves in class 350658.dw have rank $$1$$.

## Modular form 350658.2.a.dw

sage: E.q_eigenform(10)

$$q + q^{2} + q^{4} + 2q^{5} - q^{7} + q^{8} + 2q^{10} + 2q^{13} - q^{14} + q^{16} - 6q^{17} - 4q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.