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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 1342c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1342.b3 | 1342c1 | \([1, 1, 1, -13960, 629001]\) | \(300872095888141441/22515023872\) | \(22515023872\) | \([5]\) | \(3200\) | \(1.0360\) | \(\Gamma_0(N)\)-optimal |
1342.b2 | 1342c2 | \([1, 1, 1, -187880, -31262199]\) | \(733441552889589371521/4352738523915232\) | \(4352738523915232\) | \([5]\) | \(16000\) | \(1.8407\) | |
1342.b1 | 1342c3 | \([1, 1, 1, -117257250, -488766109679]\) | \(178296503348692983836197044001/1342\) | \(1342\) | \([]\) | \(80000\) | \(2.6454\) |
Rank
sage: E.rank()
The elliptic curves in class 1342c have rank \(0\).
Complex multiplication
The elliptic curves in class 1342c do not have complex multiplication.Modular form 1342.2.a.c
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.
\(\left(\begin{array}{rrr} 1 & 5 & 25 \\ 5 & 1 & 5 \\ 25 & 5 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.