# Properties

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

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

sage: E.isogeny_class()

## Elliptic curves in class 68400.ds

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
68400.ds1 68400fa4 $$[0, 0, 0, -2235675, -1284781750]$$ $$26487576322129/44531250$$ $$2077650000000000000$$ $$$$ $$1179648$$ $$2.4101$$
68400.ds2 68400fa2 $$[0, 0, 0, -183675, -6385750]$$ $$14688124849/8122500$$ $$378963360000000000$$ $$[2, 2]$$ $$589824$$ $$2.0635$$
68400.ds3 68400fa1 $$[0, 0, 0, -111675, 14278250]$$ $$3301293169/22800$$ $$1063756800000000$$ $$$$ $$294912$$ $$1.7169$$ $$\Gamma_0(N)$$-optimal
68400.ds4 68400fa3 $$[0, 0, 0, 716325, -50485750]$$ $$871257511151/527800050$$ $$-24625039132800000000$$ $$$$ $$1179648$$ $$2.4101$$

## Rank

sage: E.rank()

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

## Complex multiplication

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

## 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 LMFDB 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 LMFDB labels. 