# Properties

 Label 450g Number of curves 8 Conductor 450 CM no Rank 0 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("450.d1")

sage: E.isogeny_class()

## Elliptic curves in class 450g

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
450.d8 450g1 [1, -1, 0, 333, -7259] [2] 384 $$\Gamma_0(N)$$-optimal
450.d6 450g2 [1, -1, 0, -4167, -92759] [2, 2] 768
450.d7 450g3 [1, -1, 0, -3042, 212116] [2] 1152
450.d4 450g4 [1, -1, 0, -64917, -6350009] [2] 1536
450.d5 450g5 [1, -1, 0, -15417, 638491] [2] 1536
450.d3 450g6 [1, -1, 0, -75042, 7916116] [2, 2] 2304
450.d2 450g7 [1, -1, 0, -102042, 1733116] [2] 4608
450.d1 450g8 [1, -1, 0, -1200042, 506291116] [2] 4608

## Rank

sage: E.rank()

The elliptic curves in class 450g have rank $$0$$.

## Modular form450.2.a.d

sage: E.q_eigenform(10)

$$q - q^{2} + q^{4} + 4q^{7} - q^{8} - 2q^{13} - 4q^{14} + q^{16} + 6q^{17} - 4q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.