# Properties

 Label 25350cs Number of curves $6$ Conductor $25350$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("25350.cz1")

sage: E.isogeny_class()

## Elliptic curves in class 25350cs

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
25350.cz6 25350cs1 [1, 0, 0, 63287, 2459417] [2] 258048 $$\Gamma_0(N)$$-optimal
25350.cz5 25350cs2 [1, 0, 0, -274713, 20373417] [2, 2] 516096
25350.cz3 25350cs3 [1, 0, 0, -2387213, -1405564083] [2, 2] 1032192
25350.cz2 25350cs4 [1, 0, 0, -3570213, 2594158917] [2] 1032192
25350.cz4 25350cs5 [1, 0, 0, -485963, -3582495333] [2] 2064384
25350.cz1 25350cs6 [1, 0, 0, -38088463, -90480182833] [2] 2064384

## Rank

sage: E.rank()

The elliptic curves in class 25350cs have rank $$1$$.

## Modular form 25350.2.a.cz

sage: E.q_eigenform(10)

$$q + q^{2} + q^{3} + q^{4} + q^{6} + q^{8} + q^{9} - 4q^{11} + q^{12} + q^{16} + 6q^{17} + q^{18} - 4q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.