# Properties

 Label 51600.cs Number of curves $2$ Conductor $51600$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("cs1")

sage: E.isogeny_class()

## Elliptic curves in class 51600.cs

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
51600.cs1 51600df1 $$[0, 1, 0, -128, -972]$$ $$-2282665/2322$$ $$-237772800$$ $$[]$$ $$17280$$ $$0.30276$$ $$\Gamma_0(N)$$-optimal
51600.cs2 51600df2 $$[0, 1, 0, 1072, 16788]$$ $$1329238535/1908168$$ $$-195396403200$$ $$[]$$ $$51840$$ $$0.85207$$

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form 51600.2.a.cs

sage: E.q_eigenform(10)

$$q + q^{3} - q^{7} + q^{9} + 4q^{13} + 6q^{17} - 2q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.