# Properties

 Label 25410.ct Number of curves 8 Conductor 25410 CM no Rank 1 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("25410.ct1")

sage: E.isogeny_class()

## Elliptic curves in class 25410.ct

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
25410.ct1 25410ct8 [1, 0, 0, -42499135, 106635995225] [2] 1658880
25410.ct2 25410ct6 [1, 0, 0, -2656255, 1665943577] [2, 2] 829440
25410.ct3 25410ct7 [1, 0, 0, -2462655, 1919133657] [2] 1658880
25410.ct4 25410ct5 [1, 0, 0, -527260, 144727100] [2] 552960
25410.ct5 25410ct3 [1, 0, 0, -178175, 21985305] [2] 414720
25410.ct6 25410ct2 [1, 0, 0, -69880, -3738448] [2, 2] 276480
25410.ct7 25410ct1 [1, 0, 0, -60200, -5688000] [2] 138240 $$\Gamma_0(N)$$-optimal
25410.ct8 25410ct4 [1, 0, 0, 232620, -27393948] [2] 552960

## Rank

sage: E.rank()

The elliptic curves in class 25410.ct have rank $$1$$.

## Modular form 25410.2.a.ct

sage: E.q_eigenform(10)

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

## 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 LMFDB numbering.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.