# Properties

 Label 137904.cv Number of curves $6$ Conductor $137904$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("137904.cv1")

sage: E.isogeny_class()

## Elliptic curves in class 137904.cv

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
137904.cv1 137904p5 [0, 1, 0, -54550552, 155034754772] [4] 11010048
137904.cv2 137904p3 [0, 1, 0, -3755912, 1899074100] [2, 2] 5505024
137904.cv3 137904p2 [0, 1, 0, -1471032, -664561260] [2, 2] 2752512
137904.cv4 137904p1 [0, 1, 0, -1457512, -677762188] [2] 1376256 $$\Gamma_0(N)$$-optimal
137904.cv5 137904p4 [0, 1, 0, 597528, -2383120908] [2] 5505024
137904.cv6 137904p6 [0, 1, 0, 10480648, 12872614548] [2] 11010048

## Rank

sage: E.rank()

The elliptic curves in class 137904.cv have rank $$0$$.

## Modular form 137904.2.a.cv

sage: E.q_eigenform(10)

$$q + q^{3} + 2q^{5} + q^{9} + 4q^{11} + 2q^{15} + q^{17} - 4q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.