# Properties

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

Show commands for: SageMath
sage: E = EllipticCurve("j1")

sage: E.isogeny_class()

## Elliptic curves in class 72450.j

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
72450.j1 72450b1 $$[1, -1, 0, -286917, 58100741]$$ $$8493409990827/185150000$$ $$56942303906250000$$ $$$$ $$737280$$ $$2.0033$$ $$\Gamma_0(N)$$-optimal
72450.j2 72450b2 $$[1, -1, 0, 23583, 177022241]$$ $$4716275733/44023437500$$ $$-13539270629882812500$$ $$$$ $$1474560$$ $$2.3499$$

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form 72450.2.a.j

sage: E.q_eigenform(10)

$$q - q^{2} + q^{4} - q^{7} - q^{8} - 2q^{11} - 2q^{13} + q^{14} + q^{16} + 2q^{17} - 4q^{19} + O(q^{20})$$ ## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 