# Properties

 Label 462.a Number of curves $4$ Conductor $462$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
E = EllipticCurve("a1")

E.isogeny_class()

## Elliptic curves in class 462.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
462.a1 462c3 $$[1, 1, 0, -226, -1406]$$ $$1285429208617/614922$$ $$614922$$ $$$$ $$128$$ $$0.065661$$
462.a2 462c4 $$[1, 1, 0, -126, 486]$$ $$223980311017/4278582$$ $$4278582$$ $$$$ $$128$$ $$0.065661$$
462.a3 462c2 $$[1, 1, 0, -16, -20]$$ $$498677257/213444$$ $$213444$$ $$[2, 2]$$ $$64$$ $$-0.28091$$
462.a4 462c1 $$[1, 1, 0, 4, 0]$$ $$4657463/3696$$ $$-3696$$ $$$$ $$32$$ $$-0.62749$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 462.a have rank $$1$$.

## Complex multiplication

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

## Modular form462.2.a.a

sage: E.q_eigenform(10)

$$q - q^{2} - q^{3} + q^{4} - 2 q^{5} + q^{6} + q^{7} - q^{8} + q^{9} + 2 q^{10} + q^{11} - q^{12} + 2 q^{13} - q^{14} + 2 q^{15} + q^{16} - 6 q^{17} - q^{18} - 4 q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 