# Properties

 Label 68400ec Number of curves $3$ Conductor $68400$ CM no Rank $1$ Graph

# Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("ec1")

sage: E.isogeny_class()

## Elliptic curves in class 68400ec

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
68400.cs3 68400ec1 $$[0, 0, 0, 2400, -2000]$$ $$32768/19$$ $$-886464000000$$ $$[]$$ $$62208$$ $$0.98200$$ $$\Gamma_0(N)$$-optimal
68400.cs2 68400ec2 $$[0, 0, 0, -33600, -2522000]$$ $$-89915392/6859$$ $$-320013504000000$$ $$[]$$ $$186624$$ $$1.5313$$
68400.cs1 68400ec3 $$[0, 0, 0, -2769600, -1774082000]$$ $$-50357871050752/19$$ $$-886464000000$$ $$[]$$ $$559872$$ $$2.0806$$

## Rank

sage: E.rank()

The elliptic curves in class 68400ec have rank $$1$$.

## Complex multiplication

The elliptic curves in class 68400ec do not have complex multiplication.

## Modular form 68400.2.a.ec

sage: E.q_eigenform(10)

$$q - q^{7} + 3q^{11} + 4q^{13} - 3q^{17} - q^{19} + O(q^{20})$$

## 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 Cremona numbering.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.