# Properties

 Label 51600.y Number of curves $4$ Conductor $51600$ CM no Rank $1$ Graph # Learn more

Show commands for: SageMath
sage: E = EllipticCurve("y1")

sage: E.isogeny_class()

## Elliptic curves in class 51600.y

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
51600.y1 51600bz4 $$[0, -1, 0, -275408, -55538688]$$ $$36097320816649/80625$$ $$5160000000000$$ $$$$ $$270336$$ $$1.6845$$
51600.y2 51600bz3 $$[0, -1, 0, -47408, 2877312]$$ $$184122897769/51282015$$ $$3282048960000000$$ $$$$ $$270336$$ $$1.6845$$
51600.y3 51600bz2 $$[0, -1, 0, -17408, -842688]$$ $$9116230969/416025$$ $$26625600000000$$ $$[2, 2]$$ $$135168$$ $$1.3379$$
51600.y4 51600bz1 $$[0, -1, 0, 592, -50688]$$ $$357911/17415$$ $$-1114560000000$$ $$$$ $$67584$$ $$0.99136$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 51600.y have rank $$1$$.

## Complex multiplication

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

## Modular form 51600.2.a.y

sage: E.q_eigenform(10)

$$q - q^{3} + q^{9} - 4q^{11} - 6q^{13} + 2q^{17} + O(q^{20})$$ ## 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.

$$\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 