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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 4704.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
4704.e1 | 4704f2 | \([0, -1, 0, -1584, -23736]\) | \(7301384/3\) | \(180708864\) | \([2]\) | \(2304\) | \(0.54673\) | |
4704.e2 | 4704f3 | \([0, -1, 0, -849, 9633]\) | \(140608/3\) | \(1445670912\) | \([2]\) | \(2304\) | \(0.54673\) | |
4704.e3 | 4704f1 | \([0, -1, 0, -114, -216]\) | \(21952/9\) | \(67765824\) | \([2, 2]\) | \(1152\) | \(0.20016\) | \(\Gamma_0(N)\)-optimal |
4704.e4 | 4704f4 | \([0, -1, 0, 376, -1980]\) | \(97336/81\) | \(-4879139328\) | \([2]\) | \(2304\) | \(0.54673\) |
Rank
sage: E.rank()
The elliptic curves in class 4704.e have rank \(0\).
Complex multiplication
The elliptic curves in class 4704.e do not have complex multiplication.Modular form 4704.2.a.e
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.