# Properties

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

Show commands for: SageMath
sage: E = EllipticCurve("cm1")

sage: E.isogeny_class()

## Elliptic curves in class 72450cm

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
72450.cs1 72450cm1 $$[1, -1, 1, -1108485380, 5939475630247]$$ $$489781415227546051766883/233890092903563264000$$ $$71932167165950558208000000000$$ $$$$ $$74317824$$ $$4.2310$$ $$\Gamma_0(N)$$-optimal
72450.cs2 72450cm2 $$[1, -1, 1, 3978746620, 45172208814247]$$ $$22649115256119592694355357/15973509811739648000000$$ $$-4912603025382367056000000000000$$ $$$$ $$148635648$$ $$4.5776$$

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form 72450.2.a.cm

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 Cremona 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 Cremona labels. 