# Properties

 Label 25200ef Number of curves $8$ Conductor $25200$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("25200.ei1")

sage: E.isogeny_class()

## Elliptic curves in class 25200ef

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
25200.ei7 25200ef1 [0, 0, 0, -147675, -7685750] [2] 221184 $$\Gamma_0(N)$$-optimal
25200.ei5 25200ef2 [0, 0, 0, -1299675, 564858250] [2, 2] 442368
25200.ei4 25200ef3 [0, 0, 0, -9651675, -11541221750] [2] 663552
25200.ei6 25200ef4 [0, 0, 0, -291675, 1418634250] [2] 884736
25200.ei2 25200ef5 [0, 0, 0, -20739675, 36353898250] [4] 884736
25200.ei3 25200ef6 [0, 0, 0, -9723675, -11360285750] [2, 2] 1327104
25200.ei8 25200ef7 [0, 0, 0, 2624325, -38241881750] [2] 2654208
25200.ei1 25200ef8 [0, 0, 0, -23223675, 27101214250] [4] 2654208

## Rank

sage: E.rank()

The elliptic curves in class 25200ef have rank $$0$$.

## Modular form 25200.2.a.ei

sage: E.q_eigenform(10)

$$q + q^{7} - 2q^{13} - 6q^{17} + 4q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.