Show commands:
SageMath
E = EllipticCurve("gd1")
E.isogeny_class()
Elliptic curves in class 141120gd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 141120.lo4 | 141120gd1 | \([0, 0, 0, -25872, 1319864]\) | \(588791808/109375\) | \(355770576000000\) | \([2]\) | \(442368\) | \(1.5103\) | \(\Gamma_0(N)\)-optimal |
| 141120.lo3 | 141120gd2 | \([0, 0, 0, -393372, 94958864]\) | \(129348709488/6125\) | \(318770436096000\) | \([2]\) | \(884736\) | \(1.8568\) | |
| 141120.lo2 | 141120gd3 | \([0, 0, 0, -613872, -184997736]\) | \(10788913152/8575\) | \(20333569192473600\) | \([2]\) | \(1327104\) | \(2.0596\) | |
| 141120.lo1 | 141120gd4 | \([0, 0, 0, -746172, -99426096]\) | \(1210991472/588245\) | \(22318125545659023360\) | \([2]\) | \(2654208\) | \(2.4062\) |
Rank
sage: E.rank()
The elliptic curves in class 141120gd have rank \(2\).
Complex multiplication
The elliptic curves in class 141120gd do not have complex multiplication.Modular form 141120.2.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 Cremona 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 Cremona labels.
