# Properties

 Label 303450.gp Number of curves $8$ Conductor $303450$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("303450.gp1")

sage: E.isogeny_class()

## Elliptic curves in class 303450.gp

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
303450.gp1 303450gp8 [1, 0, 0, -2537654963, -49203821951583] [2] 127401984
303450.gp2 303450gp6 [1, 0, 0, -158606963, -768783719583] [2, 2] 63700992
303450.gp3 303450gp7 [1, 0, 0, -147046963, -885597519583] [2] 127401984
303450.gp4 303450gp5 [1, 0, 0, -31483088, -66800154708] [2] 42467328
303450.gp5 303450gp3 [1, 0, 0, -10638963, -10151783583] [2] 31850496
303450.gp6 303450gp2 [1, 0, 0, -4172588, 1721889792] [2, 2] 21233664
303450.gp7 303450gp1 [1, 0, 0, -3594588, 2621835792] [2] 10616832 $$\Gamma_0(N)$$-optimal
303450.gp8 303450gp4 [1, 0, 0, 13889912, 12649702292] [2] 42467328

## Rank

sage: E.rank()

The elliptic curves in class 303450.gp have rank $$0$$.

## Modular form 303450.2.a.gp

sage: E.q_eigenform(10)

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

## 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 LMFDB numbering.

$$\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 3 & 4 & 6 & 12 & 12 \\ 2 & 1 & 2 & 6 & 2 & 3 & 6 & 6 \\ 4 & 2 & 1 & 12 & 4 & 6 & 12 & 3 \\ 3 & 6 & 12 & 1 & 12 & 2 & 4 & 4 \\ 4 & 2 & 4 & 12 & 1 & 6 & 3 & 12 \\ 6 & 3 & 6 & 2 & 6 & 1 & 2 & 2 \\ 12 & 6 & 12 & 4 & 3 & 2 & 1 & 4 \\ 12 & 6 & 3 & 4 & 12 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.