# Properties

 Label 25200.i Number of curves $6$ Conductor $25200$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("25200.i1")

sage: E.isogeny_class()

## Elliptic curves in class 25200.i

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
25200.i1 25200eb4 [0, 0, 0, -4838475, 4096480250] [2] 393216
25200.i2 25200eb6 [0, 0, 0, -3290475, -2275339750] [2] 786432
25200.i3 25200eb3 [0, 0, 0, -374475, 31216250] [2, 2] 393216
25200.i4 25200eb2 [0, 0, 0, -302475, 63976250] [2, 2] 196608
25200.i5 25200eb1 [0, 0, 0, -14475, 1480250] [2] 98304 $$\Gamma_0(N)$$-optimal
25200.i6 25200eb5 [0, 0, 0, 1389525, 241132250] [2] 786432

## Rank

sage: E.rank()

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

## Modular form 25200.2.a.i

sage: E.q_eigenform(10)

$$q - q^{7} - 4q^{11} - 6q^{13} + 2q^{17} + 4q^{19} + O(q^{20})$$

## 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 LMFDB numbering.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.