# Properties

 Label 225318e Number of curves 6 Conductor 225318 CM no Rank 1 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("225318.bg1")

sage: E.isogeny_class()

## Elliptic curves in class 225318e

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
225318.bg5 225318e1 [1, 0, 0, -75152, -7360512] [2] 1695744 $$\Gamma_0(N)$$-optimal
225318.bg4 225318e2 [1, 0, 0, -251872, 40106480] [2, 2] 3391488
225318.bg2 225318e3 [1, 0, 0, -3830452, 2885077580] [2, 2] 6782976
225318.bg6 225318e4 [1, 0, 0, 499188, 233729748] [2] 6782976
225318.bg1 225318e5 [1, 0, 0, -61286542, 184664655122] [2] 13565952
225318.bg3 225318e6 [1, 0, 0, -3631642, 3197964758] [2] 13565952

## Rank

sage: E.rank()

The elliptic curves in class 225318e have rank $$1$$.

## Modular form 225318.2.a.bg

sage: E.q_eigenform(10)

$$q + q^{2} + q^{3} + q^{4} + 2q^{5} + q^{6} + q^{8} + q^{9} + 2q^{10} + 4q^{11} + q^{12} + 2q^{13} + 2q^{15} + q^{16} + q^{17} + q^{18} - 4q^{19} + O(q^{20})$$

## 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 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.