# Properties

 Label 637.b Number of curves $3$ Conductor $637$ CM no Rank $0$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("b1")

sage: E.isogeny_class()

## Elliptic curves in class 637.b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
637.b1 637b3 $$[0, -1, 1, -5749, 415463]$$ $$-178643795968/524596891$$ $$-61718299629259$$ $$[]$$ $$1728$$ $$1.3331$$
637.b2 637b1 $$[0, -1, 1, -359, -2507]$$ $$-43614208/91$$ $$-10706059$$ $$[]$$ $$192$$ $$0.23445$$ $$\Gamma_0(N)$$-optimal
637.b3 637b2 $$[0, -1, 1, 621, -13238]$$ $$224755712/753571$$ $$-88656874579$$ $$[]$$ $$576$$ $$0.78375$$

## Rank

sage: E.rank()

The elliptic curves in class 637.b have rank $$0$$.

## Complex multiplication

The elliptic curves in class 637.b do not have complex multiplication.

## Modular form637.2.a.b

sage: E.q_eigenform(10)

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

## 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.

$$\left(\begin{array}{rrr} 1 & 9 & 3 \\ 9 & 1 & 3 \\ 3 & 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 