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SageMath
E = EllipticCurve("m1")
E.isogeny_class()
Elliptic curves in class 67158.m
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 67158.m1 | 67158w1 | \([1, -1, 0, -140967, -20336643]\) | \(-424962187484640625/6846414848\) | \(-4991036424192\) | \([]\) | \(336960\) | \(1.5693\) | \(\Gamma_0(N)\)-optimal |
| 67158.m2 | 67158w2 | \([1, -1, 0, -57447, -44167491]\) | \(-28760901707616625/1140118101991232\) | \(-831146096351608128\) | \([3]\) | \(1010880\) | \(2.1186\) | |
| 67158.m3 | 67158w3 | \([1, -1, 0, 515853, 1177718265]\) | \(20824452493149863375/834169168835812628\) | \(-608109324081307405812\) | \([3]\) | \(3032640\) | \(2.6679\) |
Rank
sage: E.rank()
The elliptic curves in class 67158.m have rank \(1\).
Complex multiplication
The elliptic curves in class 67158.m do not have complex multiplication.Modular form 67158.2.a.m
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.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
