# Properties

 Label 127050bf Number of curves 8 Conductor 127050 CM no Rank 1 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("127050.bs1")

sage: E.isogeny_class()

## Elliptic curves in class 127050bf

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
127050.bs7 127050bf1 [1, 1, 0, -1505000, -711000000] [2] 3317760 $$\Gamma_0(N)$$-optimal
127050.bs6 127050bf2 [1, 1, 0, -1747000, -467306000] [2, 2] 6635520
127050.bs5 127050bf3 [1, 1, 0, -4454375, 2748163125] [2] 9953280
127050.bs8 127050bf4 [1, 1, 0, 5815500, -3424243500] [2] 13271040
127050.bs4 127050bf5 [1, 1, 0, -13181500, 18090887500] [2] 13271040
127050.bs2 127050bf6 [1, 1, 0, -66406375, 208242947125] [2, 2] 19906560
127050.bs3 127050bf7 [1, 1, 0, -61566375, 239891707125] [2] 39813120
127050.bs1 127050bf8 [1, 1, 0, -1062478375, 13329499403125] [2] 39813120

## Rank

sage: E.rank()

The elliptic curves in class 127050bf have rank $$1$$.

## Modular form 127050.2.a.bs

sage: E.q_eigenform(10)

$$q - q^{2} - q^{3} + q^{4} + q^{6} + q^{7} - q^{8} + q^{9} - q^{12} + 2q^{13} - q^{14} + q^{16} - 6q^{17} - q^{18} - 8q^{19} + O(q^{20})$$

## 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.