# Properties

 Label 265200ds Number of curves 4 Conductor 265200 CM no Rank 1 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("265200.ds1")

sage: E.isogeny_class()

## Elliptic curves in class 265200ds

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
265200.ds4 265200ds1 [0, -1, 0, 631992, -33001488] [2] 6635520 $$\Gamma_0(N)$$-optimal
265200.ds3 265200ds2 [0, -1, 0, -2568008, -263401488] [2, 2] 13271040
265200.ds2 265200ds3 [0, -1, 0, -25688008, 49860758512] [2] 26542080
265200.ds1 265200ds4 [0, -1, 0, -30648008, -65184361488] [2] 26542080

## Rank

sage: E.rank()

The elliptic curves in class 265200ds have rank $$1$$.

## Modular form 265200.2.a.ds

sage: E.q_eigenform(10)

$$q - q^{3} + 4q^{7} + q^{9} + 4q^{11} - q^{13} + q^{17} - 8q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.