# Properties

 Label 25200cw Number of curves $4$ Conductor $25200$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("25200.ek1")

sage: E.isogeny_class()

## Elliptic curves in class 25200cw

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
25200.ek2 25200cw1 [0, 0, 0, -42075, -3317750]  55296 $$\Gamma_0(N)$$-optimal
25200.ek3 25200cw2 [0, 0, 0, -30075, -5249750]  110592
25200.ek1 25200cw3 [0, 0, 0, -168075, 23240250]  165888
25200.ek4 25200cw4 [0, 0, 0, 263925, 123032250]  331776

## Rank

sage: E.rank()

The elliptic curves in class 25200cw have rank $$1$$.

## Modular form 25200.2.a.ek

sage: E.q_eigenform(10)

$$q + q^{7} - 2q^{13} - 2q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels. 