# Properties

 Label 525b Number of curves 6 Conductor 525 CM no Rank 0 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("525.d1")

sage: E.isogeny_class()

## Elliptic curves in class 525b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
525.d6 525b1 [1, 1, 0, 25, 0] [2] 64 $$\Gamma_0(N)$$-optimal
525.d5 525b2 [1, 1, 0, -100, -125] [2, 2] 128
525.d2 525b3 [1, 1, 0, -1225, -17000] [2, 2] 256
525.d3 525b4 [1, 1, 0, -975, 11250] [2] 256
525.d1 525b5 [1, 1, 0, -19600, -1064375] [2] 512
525.d4 525b6 [1, 1, 0, -850, -27125] [2] 512

## Rank

sage: E.rank()

The elliptic curves in class 525b have rank $$0$$.

## Modular form525.2.a.d

sage: E.q_eigenform(10)

$$q + q^{2} - q^{3} - q^{4} - q^{6} + q^{7} - 3q^{8} + q^{9} + 4q^{11} + q^{12} + 2q^{13} + q^{14} - q^{16} + 6q^{17} + q^{18} + 4q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.