# Properties

 Label 450.f Number of curves $4$ Conductor $450$ CM no Rank $0$ Graph

# Related objects

Show commands: SageMath
sage: E = EllipticCurve("f1")

sage: E.isogeny_class()

## Elliptic curves in class 450.f

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
450.f1 450a4 $$[1, -1, 1, -186305, -30804303]$$ $$502270291349/1889568$$ $$2690420062500000$$ $$[2]$$ $$3200$$ $$1.8200$$
450.f2 450a2 $$[1, -1, 1, -11930, 504447]$$ $$131872229/18$$ $$25628906250$$ $$[2]$$ $$640$$ $$1.0153$$
450.f3 450a3 $$[1, -1, 1, -6305, -924303]$$ $$-19465109/248832$$ $$-354294000000000$$ $$[2]$$ $$1600$$ $$1.4735$$
450.f4 450a1 $$[1, -1, 1, -680, 9447]$$ $$-24389/12$$ $$-17085937500$$ $$[2]$$ $$320$$ $$0.66875$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

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

## Complex multiplication

The elliptic curves in class 450.f do not have complex multiplication.

## Modular form450.2.a.f

sage: E.q_eigenform(10)

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

## 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.