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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 858a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 858.a4 | 858a1 | \([1, 1, 0, 6, -108]\) | \(18191447/5271552\) | \(-5271552\) | \([2]\) | \(192\) | \(-0.031021\) | \(\Gamma_0(N)\)-optimal |
| 858.a3 | 858a2 | \([1, 1, 0, -314, -2220]\) | \(3440899317673/106007616\) | \(106007616\) | \([2, 2]\) | \(384\) | \(0.31555\) | |
| 858.a1 | 858a3 | \([1, 1, 0, -4994, -137940]\) | \(13778603383488553/13703976\) | \(13703976\) | \([2]\) | \(768\) | \(0.66213\) | |
| 858.a2 | 858a4 | \([1, 1, 0, -754, 4732]\) | \(47504791830313/16490207448\) | \(16490207448\) | \([2]\) | \(768\) | \(0.66213\) |
Rank
sage: E.rank()
The elliptic curves in class 858a have rank \(0\).
Complex multiplication
The elliptic curves in class 858a do not have complex multiplication.Modular form 858.2.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.
