# Properties

 Label 3570.w Number of curves 8 Conductor 3570 CM no Rank 0 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("3570.w1")
sage: E.isogeny_class()

## Elliptic curves in class 3570.w

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
3570.w1 3570v7 [1, 0, 0, -13299866, -18656185104] 2 221184
3570.w2 3570v8 [1, 0, 0, -8734866, 9830443896] 2 221184
3570.w3 3570v5 [1, 0, 0, -8709126, 9891863460] 6 73728
3570.w4 3570v6 [1, 0, 0, -1017366, -151370604] 4 110592
3570.w5 3570v4 [1, 0, 0, -571526, 138219300] 6 73728
3570.w6 3570v2 [1, 0, 0, -544326, 154522980] 12 36864
3570.w7 3570v1 [1, 0, 0, -32326, 2663780] 6 18432 $$\Gamma_0(N)$$-optimal
3570.w8 3570v3 [1, 0, 0, 232634, -18120604] 2 55296

## Rank

sage: E.rank()

The elliptic curves in class 3570.w have rank $$0$$.

## Modular form3570.2.a.w

sage: E.q_eigenform(10)
$$q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + q^{7} + q^{8} + q^{9} - q^{10} + q^{12} + 2q^{13} + q^{14} - q^{15} + q^{16} - q^{17} + q^{18} - 4q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrrrrrr} 1 & 4 & 12 & 2 & 3 & 6 & 12 & 4 \\ 4 & 1 & 3 & 2 & 12 & 6 & 12 & 4 \\ 12 & 3 & 1 & 6 & 4 & 2 & 4 & 12 \\ 2 & 2 & 6 & 1 & 6 & 3 & 6 & 2 \\ 3 & 12 & 4 & 6 & 1 & 2 & 4 & 12 \\ 6 & 6 & 2 & 3 & 2 & 1 & 2 & 6 \\ 12 & 12 & 4 & 6 & 4 & 2 & 1 & 3 \\ 4 & 4 & 12 & 2 & 12 & 6 & 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.