Show commands:
SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 70a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
70.a4 | 70a1 | \([1, -1, 1, 2, -3]\) | \(1367631/2800\) | \(-2800\) | \([4]\) | \(4\) | \(-0.65416\) | \(\Gamma_0(N)\)-optimal |
70.a3 | 70a2 | \([1, -1, 1, -18, -19]\) | \(611960049/122500\) | \(122500\) | \([2, 2]\) | \(8\) | \(-0.30759\) | |
70.a1 | 70a3 | \([1, -1, 1, -268, -1619]\) | \(2121328796049/120050\) | \(120050\) | \([2]\) | \(16\) | \(0.038988\) | |
70.a2 | 70a4 | \([1, -1, 1, -88, 317]\) | \(74565301329/5468750\) | \(5468750\) | \([2]\) | \(16\) | \(0.038988\) |
Rank
sage: E.rank()
The elliptic curves in class 70a have rank \(0\).
Complex multiplication
The elliptic curves in class 70a do not have complex multiplication.Modular form 70.2.a.a
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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.