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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 242592f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 242592.f3 | 242592f1 | \([0, -1, 0, -5174, 133824]\) | \(5088448/441\) | \(1327822945344\) | \([2, 2]\) | \(442368\) | \(1.0670\) | \(\Gamma_0(N)\)-optimal |
| 242592.f1 | 242592f2 | \([0, -1, 0, -80984, 8897460]\) | \(2438569736/21\) | \(505837312512\) | \([2]\) | \(884736\) | \(1.4136\) | |
| 242592.f4 | 242592f3 | \([0, -1, 0, 5656, 610344]\) | \(830584/7203\) | \(-173502198191616\) | \([2]\) | \(884736\) | \(1.4136\) | |
| 242592.f2 | 242592f4 | \([0, -1, 0, -17809, -758207]\) | \(3241792/567\) | \(109260859502592\) | \([2]\) | \(884736\) | \(1.4136\) |
Rank
sage: E.rank()
The elliptic curves in class 242592f have rank \(0\).
Complex multiplication
The elliptic curves in class 242592f do not have complex multiplication.Modular form 242592.2.a.f
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 2 & 2 \\ 2 & 1 & 4 & 4 \\ 2 & 4 & 1 & 4 \\ 2 & 4 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
