# Properties

 Label 72450.p Number of curves $4$ Conductor $72450$ CM no Rank $0$ Graph # Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("p1")

sage: E.isogeny_class()

## Elliptic curves in class 72450.p

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
72450.p1 72450u3 $$[1, -1, 0, -649692, 66451216]$$ $$2662558086295801/1374177967680$$ $$15652745913105000000$$ $$$$ $$1658880$$ $$2.3752$$
72450.p2 72450u1 $$[1, -1, 0, -362817, -84023159]$$ $$463702796512201/15214500$$ $$173302664062500$$ $$$$ $$552960$$ $$1.8259$$ $$\Gamma_0(N)$$-optimal
72450.p3 72450u2 $$[1, -1, 0, -347067, -91661909]$$ $$-405897921250921/84358968750$$ $$-960901378417968750$$ $$$$ $$1105920$$ $$2.1724$$
72450.p4 72450u4 $$[1, -1, 0, 2437308, 514066216]$$ $$140574743422291079/91397357868600$$ $$-1041073029472021875000$$ $$$$ $$3317760$$ $$2.7218$$

## Rank

sage: E.rank()

The elliptic curves in class 72450.p have rank $$0$$.

## Complex multiplication

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

## Modular form 72450.2.a.p

sage: E.q_eigenform(10)

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

$$\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 