# Properties

 Label 61710.da Number of curves $8$ Conductor $61710$ CM no Rank $0$ Graph

# Related objects

Show commands: SageMath
sage: E = EllipticCurve("da1")

sage: E.isogeny_class()

## Elliptic curves in class 61710.da

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
61710.da1 61710cz8 $$[1, 0, 0, -24349614385, 1462465543989737]$$ $$901247067798311192691198986281/552431869440$$ $$978666755056995840$$ $$[2]$$ $$79626240$$ $$4.1597$$
61710.da2 61710cz7 $$[1, 0, 0, -1532073265, 22528378613225]$$ $$224494757451893010998773801/6152490825146276160000$$ $$10899512798686962140285760000$$ $$[2]$$ $$79626240$$ $$4.1597$$
61710.da3 61710cz6 $$[1, 0, 0, -1521851185, 22850919992297]$$ $$220031146443748723000125481/172266701724057600$$ $$305180970372973205913600$$ $$[2, 2]$$ $$39813120$$ $$3.8132$$
61710.da4 61710cz5 $$[1, 0, 0, -300673810, 2005244394572]$$ $$1696892787277117093383481/1440538624914939000$$ $$2552002046892934249779000$$ $$[2]$$ $$26542080$$ $$3.6104$$
61710.da5 61710cz4 $$[1, 0, 0, -196913890, -1052215504900]$$ $$476646772170172569823801/5862293314453125000$$ $$10385410206445892578125000$$ $$[2]$$ $$26542080$$ $$3.6104$$
61710.da6 61710cz3 $$[1, 0, 0, -94477105, 362070412265]$$ $$-52643812360427830814761/1504091705903677440$$ $$-2664590206602424709283840$$ $$[4]$$ $$19906560$$ $$3.4666$$
61710.da7 61710cz2 $$[1, 0, 0, -22978810, 16337265572]$$ $$757443433548897303481/373234243041000000$$ $$661207228835957001000000$$ $$[2, 2]$$ $$13271040$$ $$3.2639$$
61710.da8 61710cz1 $$[1, 0, 0, 5248070, 1958492900]$$ $$9023321954633914439/6156756739584000$$ $$-10907070126334170624000$$ $$[4]$$ $$6635520$$ $$2.9173$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 61710.da have rank $$0$$.

## Complex multiplication

The elliptic curves in class 61710.da do not have complex multiplication.

## Modular form 61710.2.a.da

sage: E.q_eigenform(10)

$$q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} + 4 q^{7} + q^{8} + q^{9} + q^{10} + q^{12} - 2 q^{13} + 4 q^{14} + q^{15} + q^{16} + q^{17} + q^{18} + 4 q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrrrrrr} 1 & 4 & 2 & 3 & 12 & 4 & 6 & 12 \\ 4 & 1 & 2 & 12 & 3 & 4 & 6 & 12 \\ 2 & 2 & 1 & 6 & 6 & 2 & 3 & 6 \\ 3 & 12 & 6 & 1 & 4 & 12 & 2 & 4 \\ 12 & 3 & 6 & 4 & 1 & 12 & 2 & 4 \\ 4 & 4 & 2 & 12 & 12 & 1 & 6 & 3 \\ 6 & 6 & 3 & 2 & 2 & 6 & 1 & 2 \\ 12 & 12 & 6 & 4 & 4 & 3 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.