# Properties

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

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("by1")

sage: E.isogeny_class()

## Elliptic curves in class 51600.by

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
51600.by1 51600bv1 $$[0, -1, 0, -9008, -319488]$$ $$1263214441/29025$$ $$1857600000000$$ $$[2]$$ $$110592$$ $$1.1405$$ $$\Gamma_0(N)$$-optimal
51600.by2 51600bv2 $$[0, -1, 0, 992, -999488]$$ $$1685159/6739605$$ $$-431334720000000$$ $$[2]$$ $$221184$$ $$1.4871$$

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form 51600.2.a.by

sage: E.q_eigenform(10)

$$q - q^{3} + 4q^{7} + q^{9} + 2q^{11} - 2q^{13} - 6q^{19} + O(q^{20})$$

## 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.