# Properties

 Label 5610.c Number of curves $4$ Conductor $5610$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
E = EllipticCurve("c1")

E.isogeny_class()

## Elliptic curves in class 5610.c

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
5610.c1 5610a4 $$[1, 1, 0, -11968, 498988]$$ $$189602977175292169/1402500$$ $$1402500$$ $$$$ $$8192$$ $$0.77449$$
5610.c2 5610a3 $$[1, 1, 0, -1048, 532]$$ $$127483771761289/73369857660$$ $$73369857660$$ $$$$ $$8192$$ $$0.77449$$
5610.c3 5610a2 $$[1, 1, 0, -748, 7552]$$ $$46380496070089/125888400$$ $$125888400$$ $$[2, 2]$$ $$4096$$ $$0.42792$$
5610.c4 5610a1 $$[1, 1, 0, -28, 208]$$ $$-2565726409/19388160$$ $$-19388160$$ $$$$ $$2048$$ $$0.081343$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 5610.c have rank $$1$$.

## Complex multiplication

The elliptic curves in class 5610.c do not have complex multiplication.

## Modular form5610.2.a.c

sage: E.q_eigenform(10)

$$q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} - q^{8} + q^{9} + q^{10} - q^{11} - q^{12} - 6 q^{13} + q^{15} + q^{16} - q^{17} - q^{18} + 8 q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 