# Properties

 Label 35490x Number of curves 8 Conductor 35490 CM no Rank 0 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("35490.z1")

sage: E.isogeny_class()

## Elliptic curves in class 35490x

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
35490.z7 35490x1 [1, 0, 1, -4347529, 3470020556] [2] 1548288 $$\Gamma_0(N)$$-optimal
35490.z6 35490x2 [1, 0, 1, -6997449, -1260616628] [2, 2] 3096576
35490.z5 35490x3 [1, 0, 1, -26858329, -51243878164] [2] 4644864
35490.z8 35490x4 [1, 0, 1, 27498831, -9995074724] [2] 6193152
35490.z4 35490x5 [1, 0, 1, -83892449, -295276338628] [2] 6193152
35490.z2 35490x6 [1, 0, 1, -424511949, -3366561638828] [2, 2] 9289728
35490.z3 35490x7 [1, 0, 1, -419294919, -3453337542224] [2] 18579456
35490.z1 35490x8 [1, 0, 1, -6792186899, -215458531803448] [2] 18579456

## Rank

sage: E.rank()

The elliptic curves in class 35490x have rank $$0$$.

## Modular form 35490.2.a.z

sage: E.q_eigenform(10)

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

## 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 Cremona numbering.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.