# Properties

 Label 51600.dn Number of curves $4$ Conductor $51600$ CM no Rank $0$ Graph # Learn more

Show commands for: SageMath
sage: E = EllipticCurve("dn1")

sage: E.isogeny_class()

## Elliptic curves in class 51600.dn

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
51600.dn1 51600dh3 $$[0, 1, 0, -13722408, 8494795188]$$ $$4465136636671380769/2096375976562500$$ $$134168062500000000000000$$ $$$$ $$4976640$$ $$3.1321$$
51600.dn2 51600dh1 $$[0, 1, 0, -7026408, -7170820812]$$ $$599437478278595809/33854760000$$ $$2166704640000000000$$ $$$$ $$1658880$$ $$2.5828$$ $$\Gamma_0(N)$$-optimal
51600.dn3 51600dh2 $$[0, 1, 0, -6626408, -8022820812]$$ $$-502780379797811809/143268096832200$$ $$-9169158197260800000000$$ $$$$ $$3317760$$ $$2.9293$$
51600.dn4 51600dh4 $$[0, 1, 0, 48777592, 64369795188]$$ $$200541749524551119231/144008551960031250$$ $$-9216547325442000000000000$$ $$$$ $$9953280$$ $$3.4786$$

## Rank

sage: E.rank()

The elliptic curves in class 51600.dn have rank $$0$$.

## Complex multiplication

The elliptic curves in class 51600.dn do not have complex multiplication.

## Modular form 51600.2.a.dn

sage: E.q_eigenform(10)

$$q + q^{3} + 2q^{7} + q^{9} - 2q^{13} + 6q^{17} - 8q^{19} + O(q^{20})$$ ## 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.

$$\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 