# Properties

 Label 137904bu Number of curves 6 Conductor 137904 CM no Rank 1 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("137904.bh1")

sage: E.isogeny_class()

## Elliptic curves in class 137904bu

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
137904.bh5 137904bu1 [0, -1, 0, -91992, -9929232] [2] 884736 $$\Gamma_0(N)$$-optimal
137904.bh4 137904bu2 [0, -1, 0, -308312, 54447600] [2, 2] 1769472
137904.bh2 137904bu3 [0, -1, 0, -4688792, 3909270000] [2, 2] 3538944
137904.bh6 137904bu4 [0, -1, 0, 611048, 316281328] [2] 3538944
137904.bh1 137904bu5 [0, -1, 0, -75019832, 250124174832] [2] 7077888
137904.bh3 137904bu6 [0, -1, 0, -4445432, 4332911088] [2] 7077888

## Rank

sage: E.rank()

The elliptic curves in class 137904bu have rank $$1$$.

## Modular form 137904.2.a.bh

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 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.