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SageMath
E = EllipticCurve("hd1")
E.isogeny_class()
Elliptic curves in class 169050.hd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 169050.hd1 | 169050w3 | \([1, 0, 0, -602654088, 3236266119792]\) | \(13167998447866683762601/5158996582031250000\) | \(9483606076240539550781250000\) | \([2]\) | \(141557760\) | \(4.0669\) | |
| 169050.hd2 | 169050w2 | \([1, 0, 0, -271316088, -1684434518208]\) | \(1201550658189465626281/28577902500000000\) | \(52533775800351562500000000\) | \([2, 2]\) | \(70778880\) | \(3.7203\) | |
| 169050.hd3 | 169050w1 | \([1, 0, 0, -269748088, -1705262262208]\) | \(1180838681727016392361/692428800000\) | \(1272868060800000000000\) | \([2]\) | \(35389440\) | \(3.3738\) | \(\Gamma_0(N)\)-optimal |
| 169050.hd4 | 169050w4 | \([1, 0, 0, 34933912, -5272153268208]\) | \(2564821295690373719/6533572090396050000\) | \(-12010440982234451350781250000\) | \([2]\) | \(141557760\) | \(4.0669\) |
Rank
sage: E.rank()
The elliptic curves in class 169050.hd have rank \(0\).
Complex multiplication
The elliptic curves in class 169050.hd do not have complex multiplication.Modular form 169050.2.a.hd
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
