# Properties

 Label 303450.cx Number of curves $8$ Conductor $303450$ CM no Rank $1$ Graph

# Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("303450.cx1")

sage: E.isogeny_class()

## Elliptic curves in class 303450.cx

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
303450.cx1 303450cx7 [1, 0, 1, -46608626, 77049510398] [2] 63700992
303450.cx2 303450cx4 [1, 0, 1, -41623376, 103356891398] [2] 21233664
303450.cx3 303450cx6 [1, 0, 1, -19514876, -32300864602] [2, 2] 31850496
303450.cx4 303450cx3 [1, 0, 1, -19370376, -32815284602] [2] 15925248
303450.cx5 303450cx2 [1, 0, 1, -2608376, 1605771398] [2, 2] 10616832
303450.cx6 303450cx5 [1, 0, 1, -585376, 4033371398] [2] 21233664
303450.cx7 303450cx1 [1, 0, 1, -296376, -21876602] [2] 5308416 $$\Gamma_0(N)$$-optimal
303450.cx8 303450cx8 [1, 0, 1, 5266874, -108727781602] [2] 63700992

## Rank

sage: E.rank()

The elliptic curves in class 303450.cx have rank $$1$$.

## Modular form 303450.2.a.cx

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.