# Properties

 Label 51600.z Number of curves $2$ Conductor $51600$ CM no Rank $0$ Graph # Learn more

Show commands for: SageMath
sage: E = EllipticCurve("z1")

sage: E.isogeny_class()

## Elliptic curves in class 51600.z

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
51600.z1 51600q2 $$[0, -1, 0, -6208, 100912]$$ $$13231796/5547$$ $$11094000000000$$ $$$$ $$92160$$ $$1.2001$$
51600.z2 51600q1 $$[0, -1, 0, 1292, 10912]$$ $$476656/387$$ $$-193500000000$$ $$$$ $$46080$$ $$0.85354$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form 51600.2.a.z

sage: E.q_eigenform(10)

$$q - q^{3} + q^{9} - 4q^{11} - 6q^{17} + 2q^{19} + O(q^{20})$$ ## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 