# Properties

 Label 35490cy Number of curves 8 Conductor 35490 CM no Rank 1 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("35490.cz1")

sage: E.isogeny_class()

## Elliptic curves in class 35490cy

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
35490.cz7 35490cy1 [1, 0, 0, -84081, 9373545] [2] 221184 $$\Gamma_0(N)$$-optimal
35490.cz6 35490cy2 [1, 0, 0, -97601, 6153081] [2, 2] 442368
35490.cz5 35490cy3 [1, 0, 0, -248856, -36335040] [2] 663552
35490.cz8 35490cy4 [1, 0, 0, 324899, 45276581] [2] 884736
35490.cz4 35490cy5 [1, 0, 0, -736421, -239026035] [2] 884736
35490.cz2 35490cy6 [1, 0, 0, -3709976, -2750545344] [2, 2] 1327104
35490.cz3 35490cy7 [1, 0, 0, -3439576, -3168421504] [2] 2654208
35490.cz1 35490cy8 [1, 0, 0, -59358296, -176028284160] [2] 2654208

## Rank

sage: E.rank()

The elliptic curves in class 35490cy have rank $$1$$.

## Modular form 35490.2.a.cz

sage: E.q_eigenform(10)

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

## 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.