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SageMath
E = EllipticCurve("ml1")
E.isogeny_class()
Elliptic curves in class 463680ml
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
463680.ml2 | 463680ml1 | \([0, 0, 0, -928812, -344530384]\) | \(463702796512201/15214500\) | \(2907536228352000\) | \([2]\) | \(4423680\) | \(2.0609\) | \(\Gamma_0(N)\)-optimal* |
463680.ml3 | 463680ml2 | \([0, 0, 0, -888492, -375802576]\) | \(-405897921250921/84358968750\) | \(-16121249980416000000\) | \([2]\) | \(8847360\) | \(2.4075\) | |
463680.ml1 | 463680ml3 | \([0, 0, 0, -1663212, 271518896]\) | \(2662558086295801/1374177967680\) | \(262609499177279815680\) | \([2]\) | \(13271040\) | \(2.6102\) | \(\Gamma_0(N)\)-optimal* |
463680.ml4 | 463680ml4 | \([0, 0, 0, 6239508, 2108111024]\) | \(140574743422291079/91397357868600\) | \(-17466307087226476953600\) | \([2]\) | \(26542080\) | \(2.9568\) |
Rank
sage: E.rank()
The elliptic curves in class 463680ml have rank \(1\).
Complex multiplication
The elliptic curves in class 463680ml do not have complex multiplication.Modular form 463680.2.a.ml
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.