# Properties

 Label 637.d Number of curves $2$ Conductor $637$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("d1")

sage: E.isogeny_class()

## Elliptic curves in class 637.d

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
637.d1 637c1 $$[1, -1, 0, -5252, -145223]$$ $$-56723625/13$$ $$-3672178237$$ $$[]$$ $$420$$ $$0.82691$$ $$\Gamma_0(N)$$-optimal
637.d2 637c2 $$[1, -1, 0, 30763, 6051758]$$ $$11397810375/62748517$$ $$-17724902963955733$$ $$[]$$ $$2940$$ $$1.7999$$

## Rank

sage: E.rank()

The elliptic curves in class 637.d have rank $$1$$.

## Complex multiplication

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

## Modular form637.2.a.d

sage: E.q_eigenform(10)

$$q + q^{2} - q^{4} - 3 q^{8} - 3 q^{9} - 3 q^{11} + q^{13} - q^{16} - 7 q^{17} - 3 q^{18} + 7 q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 