# Properties

 Label 301530.cs Number of curves $4$ Conductor $301530$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("301530.cs1")

sage: E.isogeny_class()

## Elliptic curves in class 301530.cs

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
301530.cs1 301530cs4 [1, 0, 0, -1608171, -785092089]  4325376
301530.cs2 301530cs3 [1, 0, 0, -116391, -8141325]  4325376
301530.cs3 301530cs2 [1, 0, 0, -100521, -12270699] [2, 2] 2162688
301530.cs4 301530cs1 [1, 0, 0, -5301, -253935]  1081344 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 301530.cs have rank $$1$$.

## Modular form 301530.2.a.cs

sage: E.q_eigenform(10)

$$q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} - 4q^{7} + q^{8} + q^{9} - q^{10} + 4q^{11} + q^{12} - 2q^{13} - 4q^{14} - q^{15} + q^{16} + 2q^{17} + q^{18} + q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 