# Properties

 Label 8712.u Number of curves $6$ Conductor $8712$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("u1")

sage: E.isogeny_class()

## Elliptic curves in class 8712.u

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
8712.u1 8712y5 $$[0, 0, 0, -418539, -104219962]$$ $$3065617154/9$$ $$23804337604608$$ $$[2]$$ $$40960$$ $$1.7960$$
8712.u2 8712y4 $$[0, 0, 0, -70059, 7136822]$$ $$28756228/3$$ $$3967389600768$$ $$[2]$$ $$20480$$ $$1.4495$$
8712.u3 8712y3 $$[0, 0, 0, -26499, -1583890]$$ $$1556068/81$$ $$107119519220736$$ $$[2, 2]$$ $$20480$$ $$1.4495$$
8712.u4 8712y2 $$[0, 0, 0, -4719, 93170]$$ $$35152/9$$ $$2975542200576$$ $$[2, 2]$$ $$10240$$ $$1.1029$$
8712.u5 8712y1 $$[0, 0, 0, 726, 9317]$$ $$2048/3$$ $$-61990462512$$ $$[2]$$ $$5120$$ $$0.75633$$ $$\Gamma_0(N)$$-optimal
8712.u6 8712y6 $$[0, 0, 0, 17061, -6279658]$$ $$207646/6561$$ $$-17353362113759232$$ $$[2]$$ $$40960$$ $$1.7960$$

## Rank

sage: E.rank()

The elliptic curves in class 8712.u have rank $$0$$.

## Complex multiplication

The elliptic curves in class 8712.u do not have complex multiplication.

## Modular form8712.2.a.u

sage: E.q_eigenform(10)

$$q + 2q^{5} + 2q^{13} + 2q^{17} + 4q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 8 & 4 \\ 8 & 1 & 4 & 2 & 4 & 8 \\ 2 & 4 & 1 & 2 & 4 & 2 \\ 4 & 2 & 2 & 1 & 2 & 4 \\ 8 & 4 & 4 & 2 & 1 & 8 \\ 4 & 8 & 2 & 4 & 8 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.