Properties

 Label 8664g Number of curves $6$ Conductor $8664$ CM no Rank $1$ Graph

Related objects

Show commands for: SageMath
sage: E = EllipticCurve("g1")

sage: E.isogeny_class()

Elliptic curves in class 8664g

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
8664.j5 8664g1 $$[0, 1, 0, 241, -1698]$$ $$2048/3$$ $$-2258202288$$ $$[2]$$ $$3456$$ $$0.48029$$ $$\Gamma_0(N)$$-optimal
8664.j4 8664g2 $$[0, 1, 0, -1564, -18304]$$ $$35152/9$$ $$108393709824$$ $$[2, 2]$$ $$6912$$ $$0.82687$$
8664.j2 8664g3 $$[0, 1, 0, -23224, -1369888]$$ $$28756228/3$$ $$144524946432$$ $$[2]$$ $$13824$$ $$1.1734$$
8664.j3 8664g4 $$[0, 1, 0, -8784, 299376]$$ $$1556068/81$$ $$3902173553664$$ $$[2, 2]$$ $$13824$$ $$1.1734$$
8664.j1 8664g5 $$[0, 1, 0, -138744, 19845360]$$ $$3065617154/9$$ $$867149678592$$ $$[2]$$ $$27648$$ $$1.5200$$
8664.j6 8664g6 $$[0, 1, 0, 5656, 1200432]$$ $$207646/6561$$ $$-632152115693568$$ $$[2]$$ $$27648$$ $$1.5200$$

Rank

sage: E.rank()

The elliptic curves in class 8664g have rank $$1$$.

Complex multiplication

The elliptic curves in class 8664g do not have complex multiplication.

Modular form8664.2.a.g

sage: E.q_eigenform(10)

$$q + q^{3} - 2q^{5} + q^{9} + 4q^{11} + 2q^{13} - 2q^{15} + 2q^{17} + O(q^{20})$$

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.