# Properties

 Label 10350.bg Number of curves $6$ Conductor $10350$ CM no Rank $1$ Graph

# Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("10350.bg1")

sage: E.isogeny_class()

## Elliptic curves in class 10350.bg

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
10350.bg1 10350bj3 [1, -1, 1, -24840005, -47645185503] [2] 294912
10350.bg2 10350bj5 [1, -1, 1, -5811755, 4620806997] [2] 589824
10350.bg3 10350bj4 [1, -1, 1, -1593005, -703255503] [2, 2] 294912
10350.bg4 10350bj2 [1, -1, 1, -1552505, -744160503] [2, 2] 147456
10350.bg5 10350bj1 [1, -1, 1, -94505, -12244503] [2] 73728 $$\Gamma_0(N)$$-optimal
10350.bg6 10350bj6 [1, -1, 1, 1977745, -3409884003] [2] 589824

## Rank

sage: E.rank()

The elliptic curves in class 10350.bg have rank $$1$$.

## Modular form 10350.2.a.bg

sage: E.q_eigenform(10)

$$q + q^{2} + q^{4} + q^{8} - 4q^{11} + 2q^{13} + q^{16} - 6q^{17} + 4q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.