# Properties

 Label 507.a Number of curves $4$ Conductor $507$ CM no Rank $1$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 507.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
507.a1 507c3 $$[1, 1, 1, -11749, -495058]$$ $$37159393753/1053$$ $$5082629877$$ $$$$ $$672$$ $$0.96411$$
507.a2 507c4 $$[1, 1, 1, -3299, 64670]$$ $$822656953/85683$$ $$413575475547$$ $$$$ $$672$$ $$0.96411$$
507.a3 507c2 $$[1, 1, 1, -764, -7324]$$ $$10218313/1521$$ $$7341576489$$ $$[2, 2]$$ $$336$$ $$0.61753$$
507.a4 507c1 $$[1, 1, 1, 81, -564]$$ $$12167/39$$ $$-188245551$$ $$$$ $$168$$ $$0.27096$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 507.a have rank $$1$$.

## Complex multiplication

The elliptic curves in class 507.a do not have complex multiplication.

## Modular form507.2.a.a

sage: E.q_eigenform(10)

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