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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 11858.j
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
11858.j1 | 11858i3 | \([1, -1, 0, -30615503, -65194232251]\) | \(15226621995131793/2324168\) | \(484408626286690952\) | \([2]\) | \(552960\) | \(2.7991\) | |
11858.j2 | 11858i4 | \([1, -1, 0, -3579263, 1001772085]\) | \(24331017010833/12004097336\) | \(2501922537589144342904\) | \([2]\) | \(552960\) | \(2.7991\) | |
11858.j3 | 11858i2 | \([1, -1, 0, -1919143, -1011953475]\) | \(3750606459153/45914176\) | \(9569541841745241664\) | \([2, 2]\) | \(276480\) | \(2.4525\) | |
11858.j4 | 11858i1 | \([1, -1, 0, -21863, -40925571]\) | \(-5545233/3469312\) | \(-723082264311328768\) | \([2]\) | \(138240\) | \(2.1059\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 11858.j have rank \(1\).
Complex multiplication
The elliptic curves in class 11858.j do not have complex multiplication.Modular form 11858.2.a.j
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.