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SageMath
E = EllipticCurve("gd1")
E.isogeny_class()
Elliptic curves in class 336336.gd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 336336.gd1 | 336336gd4 | \([0, 1, 0, -34026008, -76371177324]\) | \(9040834853442015625/4807849294248\) | \(2316855957991354171392\) | \([2]\) | \(23887872\) | \(3.0493\) | |
| 336336.gd2 | 336336gd3 | \([0, 1, 0, -1756568, -1622246508]\) | \(-1243857621903625/1637696668608\) | \(-789190145495296376832\) | \([2]\) | \(11943936\) | \(2.7027\) | |
| 336336.gd3 | 336336gd2 | \([0, 1, 0, -1297928, 444187764]\) | \(501796540869625/113170859802\) | \(54535940033927159808\) | \([2]\) | \(7962624\) | \(2.5000\) | |
| 336336.gd4 | 336336gd1 | \([0, 1, 0, 183832, 42927156]\) | \(1425727406375/2472321852\) | \(-1191387928846123008\) | \([2]\) | \(3981312\) | \(2.1534\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 336336.gd have rank \(0\).
Complex multiplication
The elliptic curves in class 336336.gd do not have complex multiplication.Modular form 336336.2.a.gd
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
