# Properties

 Label 139425.f Number of curves $6$ Conductor $139425$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("139425.f1")

sage: E.isogeny_class()

## Elliptic curves in class 139425.f

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
139425.f1 139425n3 [1, 1, 1, -29000488, 60099209906] [2] 5505024
139425.f2 139425n6 [1, 1, 1, -12713113, -16900020844] [2] 11010048
139425.f3 139425n4 [1, 1, 1, -2002738, 729256406] [2, 2] 5505024
139425.f4 139425n2 [1, 1, 1, -1812613, 938393906] [2, 2] 2752512
139425.f5 139425n1 [1, 1, 1, -101488, 17808656] [2] 1376256 $$\Gamma_0(N)$$-optimal
139425.f6 139425n5 [1, 1, 1, 5665637, 4977536156] [2] 11010048

## Rank

sage: E.rank()

The elliptic curves in class 139425.f have rank $$0$$.

## Modular form 139425.2.a.f

sage: E.q_eigenform(10)

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

## 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 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.