# Properties

 Label 6897.b Number of curves $2$ Conductor $6897$ CM no Rank $1$ Graph

# Related objects

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

E.isogeny_class()

## Elliptic curves in class 6897.b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
6897.b1 6897d2 $$[0, 1, 1, -3637663, 2669221561]$$ $$-3004935183806464000/2037123$$ $$-3608887659003$$ $$[]$$ $$64800$$ $$2.1591$$
6897.b2 6897d1 $$[0, 1, 1, -43963, 3810208]$$ $$-5304438784000/497763387$$ $$-881818203637107$$ $$[]$$ $$21600$$ $$1.6098$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 6897.b have rank $$1$$.

## Complex multiplication

The elliptic curves in class 6897.b do not have complex multiplication.

## Modular form6897.2.a.b

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.