# Properties

 Label 72450.f Number of curves $2$ Conductor $72450$ CM no Rank $1$ Graph # Learn more about

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

sage: E.isogeny_class()

## Elliptic curves in class 72450.f

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
72450.f1 72450m2 $$[1, -1, 0, -1523382, 724085576]$$ $$158909194494247023/5080516$$ $$12499974553500$$ $$$$ $$884736$$ $$2.0155$$
72450.f2 72450m1 $$[1, -1, 0, -95082, 11363876]$$ $$-38638468208943/219395344$$ $$-539794819494000$$ $$$$ $$442368$$ $$1.6689$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 72450.f have rank $$1$$.

## Complex multiplication

The elliptic curves in class 72450.f do not have complex multiplication.

## Modular form 72450.2.a.f

sage: E.q_eigenform(10)

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

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 