# Properties

 Label 68400fs Number of curves $4$ Conductor $68400$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("68400.fv1")

sage: E.isogeny_class()

## Elliptic curves in class 68400fs

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
68400.fv4 68400fs1 [0, 0, 0, -36075, 4500250] [2] 442368 $$\Gamma_0(N)$$-optimal
68400.fv3 68400fs2 [0, 0, 0, -684075, 217692250] [2, 2] 884736
68400.fv2 68400fs3 [0, 0, 0, -792075, 144360250] [2] 1769472
68400.fv1 68400fs4 [0, 0, 0, -10944075, 13935312250] [2] 1769472

## Rank

sage: E.rank()

The elliptic curves in class 68400fs have rank $$0$$.

## Modular form 68400.2.a.fv

sage: E.q_eigenform(10)

$$q + 4q^{7} - 4q^{11} + 2q^{13} - 2q^{17} + q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.