# Properties

 Label 5610t Number of curves $8$ Conductor $5610$ CM no Rank $0$ Graph

# Related objects

Show commands: SageMath
sage: E = EllipticCurve("t1")

sage: E.isogeny_class()

## Elliptic curves in class 5610t

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
5610.q8 5610t1 $$[1, 0, 1, 43372, -1467502]$$ $$9023321954633914439/6156756739584000$$ $$-6156756739584000$$ $$[6]$$ $$55296$$ $$1.7183$$ $$\Gamma_0(N)$$-optimal
5610.q7 5610t2 $$[1, 0, 1, -189908, -12291694]$$ $$757443433548897303481/373234243041000000$$ $$373234243041000000$$ $$[2, 6]$$ $$110592$$ $$2.0649$$
5610.q6 5610t3 $$[1, 0, 1, -780803, -272099842]$$ $$-52643812360427830814761/1504091705903677440$$ $$-1504091705903677440$$ $$[2]$$ $$165888$$ $$2.2676$$
5610.q4 5610t4 $$[1, 0, 1, -2484908, -1506795694]$$ $$1696892787277117093383481/1440538624914939000$$ $$1440538624914939000$$ $$[6]$$ $$221184$$ $$2.4115$$
5610.q5 5610t5 $$[1, 0, 1, -1627388, 790397138]$$ $$476646772170172569823801/5862293314453125000$$ $$5862293314453125000$$ $$[6]$$ $$221184$$ $$2.4115$$
5610.q3 5610t6 $$[1, 0, 1, -12577283, -17169377794]$$ $$220031146443748723000125481/172266701724057600$$ $$172266701724057600$$ $$[2, 2]$$ $$331776$$ $$2.6142$$
5610.q1 5610t7 $$[1, 0, 1, -201236483, -1098790303234]$$ $$901247067798311192691198986281/552431869440$$ $$552431869440$$ $$[2]$$ $$663552$$ $$2.9608$$
5610.q2 5610t8 $$[1, 0, 1, -12661763, -16927055362]$$ $$224494757451893010998773801/6152490825146276160000$$ $$6152490825146276160000$$ $$[2]$$ $$663552$$ $$2.9608$$

## Rank

sage: E.rank()

The elliptic curves in class 5610t have rank $$0$$.

## Complex multiplication

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

## Modular form5610.2.a.t

sage: E.q_eigenform(10)

$$q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} - 4 q^{7} - q^{8} + q^{9} - q^{10} + q^{11} + q^{12} + 2 q^{13} + 4 q^{14} + q^{15} + q^{16} - q^{17} - q^{18} - 4 q^{19} + O(q^{20})$$

## 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 Cremona numbering.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.