Show commands:
SageMath
E = EllipticCurve("g1")
E.isogeny_class()
Elliptic curves in class 770.g
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
770.g1 | 770g4 | \([1, 0, 0, -3520, -80238]\) | \(4823468134087681/30382271150\) | \(30382271150\) | \([2]\) | \(1152\) | \(0.84882\) | |
770.g2 | 770g2 | \([1, 0, 0, -270, 1612]\) | \(2177286259681/105875000\) | \(105875000\) | \([6]\) | \(384\) | \(0.29951\) | |
770.g3 | 770g3 | \([1, 0, 0, -90, -2720]\) | \(-80677568161/3131816380\) | \(-3131816380\) | \([2]\) | \(576\) | \(0.50224\) | |
770.g4 | 770g1 | \([1, 0, 0, 10, 100]\) | \(109902239/4312000\) | \(-4312000\) | \([6]\) | \(192\) | \(-0.047062\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 770.g have rank \(0\).
Complex multiplication
The elliptic curves in class 770.g do not have complex multiplication.Modular form 770.2.a.g
sage: E.q_eigenform(10)
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 2 & 6 \\ 3 & 1 & 6 & 2 \\ 2 & 6 & 1 & 3 \\ 6 & 2 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.