# Properties

 Label 462.f Number of curves $4$ Conductor $462$ CM no Rank $0$ Graph # Related objects

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

E.isogeny_class()

## Elliptic curves in class 462.f

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
462.f1 462g4 $$[1, 0, 0, -14133, -647829]$$ $$312196988566716625/25367712678$$ $$25367712678$$ $$$$ $$576$$ $$1.0416$$
462.f2 462g3 $$[1, 0, 0, -823, -11611]$$ $$-61653281712625/21875235228$$ $$-21875235228$$ $$$$ $$288$$ $$0.69498$$
462.f3 462g2 $$[1, 0, 0, -363, 1305]$$ $$5290763640625/2291573592$$ $$2291573592$$ $$$$ $$192$$ $$0.49225$$
462.f4 462g1 $$[1, 0, 0, 77, 161]$$ $$50447927375/39517632$$ $$-39517632$$ $$$$ $$96$$ $$0.14568$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 462.f have rank $$0$$.

## Complex multiplication

The elliptic curves in class 462.f do not have complex multiplication.

## Modular form462.2.a.f

sage: E.q_eigenform(10)

$$q + q^{2} + q^{3} + q^{4} + q^{6} + q^{7} + q^{8} + q^{9} - q^{11} + q^{12} + 2 q^{13} + q^{14} + q^{16} + q^{18} + 2 q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 