# Properties

 Label 68400fa Number of curves $4$ Conductor $68400$ CM no Rank $0$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("68400.ds1")

sage: E.isogeny_class()

## Elliptic curves in class 68400fa

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
68400.ds3 68400fa1 [0, 0, 0, -111675, 14278250]  294912 $$\Gamma_0(N)$$-optimal
68400.ds2 68400fa2 [0, 0, 0, -183675, -6385750] [2, 2] 589824
68400.ds4 68400fa3 [0, 0, 0, 716325, -50485750]  1179648
68400.ds1 68400fa4 [0, 0, 0, -2235675, -1284781750]  1179648

## Rank

sage: E.rank()

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

## Modular form 68400.2.a.ds

sage: E.q_eigenform(10)

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

## 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. 