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SageMath
E = EllipticCurve("hw1")
E.isogeny_class()
Elliptic curves in class 198450.hw
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 198450.hw1 | 198450v3 | \([1, -1, 1, -1319555, -583101053]\) | \(-189613868625/128\) | \(-171532242000000\) | \([]\) | \(2177280\) | \(2.0470\) | |
| 198450.hw2 | 198450v4 | \([1, -1, 1, -1043930, -833699303]\) | \(-1159088625/2097152\) | \(-227641124487168000000\) | \([]\) | \(6531840\) | \(2.5963\) | |
| 198450.hw3 | 198450v2 | \([1, -1, 1, -51680, 4751947]\) | \(-140625/8\) | \(-868381975125000\) | \([]\) | \(933120\) | \(1.6234\) | |
| 198450.hw4 | 198450v1 | \([1, -1, 1, 3445, 11197]\) | \(3375/2\) | \(-2680191281250\) | \([]\) | \(311040\) | \(1.0741\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 198450.hw have rank \(0\).
Complex multiplication
The elliptic curves in class 198450.hw do not have complex multiplication.Modular form 198450.2.a.hw
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 21 & 7 \\ 3 & 1 & 7 & 21 \\ 21 & 7 & 1 & 3 \\ 7 & 21 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
