# Properties

 Label 25350i Number of curves $8$ Conductor $25350$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("25350.d1")

sage: E.isogeny_class()

## Elliptic curves in class 25350i

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
25350.d8 25350i1 [1, 1, 0, 6250, 586500] [2] 110592 $$\Gamma_0(N)$$-optimal
25350.d6 25350i2 [1, 1, 0, -78250, 7600000] [2, 2] 221184
25350.d7 25350i3 [1, 1, 0, -57125, -17221875] [2] 331776
25350.d5 25350i4 [1, 1, 0, -289500, -51761250] [2] 442368
25350.d4 25350i5 [1, 1, 0, -1219000, 517515250] [2] 442368
25350.d3 25350i6 [1, 1, 0, -1409125, -643197875] [2, 2] 663552
25350.d1 25350i7 [1, 1, 0, -22534125, -41182072875] [2] 1327104
25350.d2 25350i8 [1, 1, 0, -1916125, -139746875] [2] 1327104

## Rank

sage: E.rank()

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

## Modular form 25350.2.a.d

sage: E.q_eigenform(10)

$$q - q^{2} - q^{3} + q^{4} + q^{6} - 4q^{7} - q^{8} + q^{9} - q^{12} + 4q^{14} + q^{16} - 6q^{17} - q^{18} + 4q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.