# Properties

 Label 25200.o Number of curves $4$ Conductor $25200$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("25200.o1")

sage: E.isogeny_class()

## Elliptic curves in class 25200.o

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
25200.o1 25200ea4 [0, 0, 0, -1344675, 600169250] [2] 294912
25200.o2 25200ea2 [0, 0, 0, -84675, 9229250] [2, 2] 147456
25200.o3 25200ea1 [0, 0, 0, -12675, -346750] [2] 73728 $$\Gamma_0(N)$$-optimal
25200.o4 25200ea3 [0, 0, 0, 23325, 31153250] [2] 294912

## Rank

sage: E.rank()

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

## Modular form 25200.2.a.o

sage: E.q_eigenform(10)

$$q - q^{7} - 4q^{11} + 2q^{13} - 6q^{17} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.