# Properties

 Label 54450.ef Number of curves $8$ Conductor $54450$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("54450.ef1")

sage: E.isogeny_class()

## Elliptic curves in class 54450.ef

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
54450.ef1 54450gf7 [1, -1, 1, -145205105, -673437860103] [2] 6635520
54450.ef2 54450gf8 [1, -1, 1, -12347105, -2269736103] [2] 6635520
54450.ef3 54450gf6 [1, -1, 1, -9080105, -10509110103] [2, 2] 3317760
54450.ef4 54450gf5 [1, -1, 1, -7854980, 8475426897] [2] 2211840
54450.ef5 54450gf4 [1, -1, 1, -1865480, -844235103] [2] 2211840
54450.ef6 54450gf2 [1, -1, 1, -504230, 124974897] [2, 2] 1105920
54450.ef7 54450gf3 [1, -1, 1, -368105, -281222103] [2] 1658880
54450.ef8 54450gf1 [1, -1, 1, 40270, 9540897] [2] 552960 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 54450.ef have rank $$0$$.

## Modular form 54450.2.a.ef

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 LMFDB numbering.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.