# Properties

 Label 68970bi Number of curves $4$ Conductor $68970$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("68970.bd1")

sage: E.isogeny_class()

## Elliptic curves in class 68970bi

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
68970.bd4 68970bi1 [1, 0, 1, -1213, -27832]  122880 $$\Gamma_0(N)$$-optimal
68970.bd3 68970bi2 [1, 0, 1, -22993, -1343344] [2, 2] 245760
68970.bd2 68970bi3 [1, 0, 1, -26623, -891772]  491520
68970.bd1 68970bi4 [1, 0, 1, -367843, -85900564]  491520

## Rank

sage: E.rank()

The elliptic curves in class 68970bi have rank $$1$$.

## Modular form 68970.2.a.bd

sage: E.q_eigenform(10)

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

## 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 Cremona numbering.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels. 