# Properties

 Label 450.b Number of curves $4$ Conductor $450$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("450.b1")

sage: E.isogeny_class()

## Elliptic curves in class 450.b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
450.b1 450c4 [1, -1, 0, -7452, -244944]  640
450.b2 450c2 [1, -1, 0, -477, 4131]  128
450.b3 450c3 [1, -1, 0, -252, -7344]  320
450.b4 450c1 [1, -1, 0, -27, 81]  64 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 450.b have rank $$1$$.

## Modular form450.2.a.b

sage: E.q_eigenform(10)

$$q - q^{2} + q^{4} - 2q^{7} - q^{8} - 2q^{11} - 6q^{13} + 2q^{14} + q^{16} + 2q^{17} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrr} 1 & 5 & 2 & 10 \\ 5 & 1 & 10 & 2 \\ 2 & 10 & 1 & 5 \\ 10 & 2 & 5 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 