# Properties

 Label 72450.w Number of curves $2$ Conductor $72450$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("w1")

sage: E.isogeny_class()

## Elliptic curves in class 72450.w

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
72450.w1 72450a1 $$[1, -1, 0, -123165042, -219939523884]$$ $$489781415227546051766883/233890092903563264000$$ $$98672382943690752000000000$$ $$$$ $$24772608$$ $$3.6817$$ $$\Gamma_0(N)$$-optimal
72450.w2 72450a2 $$[1, -1, 0, 442082958, -1673192131884]$$ $$22649115256119592694355357/15973509811739648000000$$ $$-6738824451827664000000000000$$ $$$$ $$49545216$$ $$4.0283$$

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form 72450.2.a.w

sage: E.q_eigenform(10)

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