# Properties

 Label 51600cu Number of curves $2$ Conductor $51600$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("cu1")

sage: E.isogeny_class()

## Elliptic curves in class 51600cu

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
51600.cl2 51600cu1 $$[0, 1, 0, -15008, -480012]$$ $$5841725401/1857600$$ $$118886400000000$$ $$$$ $$165888$$ $$1.4046$$ $$\Gamma_0(N)$$-optimal
51600.cl1 51600cu2 $$[0, 1, 0, -95008, 10879988]$$ $$1481933914201/53916840$$ $$3450677760000000$$ $$$$ $$331776$$ $$1.7512$$

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form 51600.2.a.cu

sage: E.q_eigenform(10)

$$q + q^{3} - 2q^{7} + q^{9} + 2q^{11} + 2q^{13} + 4q^{17} + 6q^{19} + O(q^{20})$$ ## 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 Cremona 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 Cremona labels. 