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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 81120f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 81120.b3 | 81120f1 | \([0, -1, 0, -2749686, -992178864]\) | \(7442744143086784/2927948765625\) | \(904489565021289000000\) | \([2, 2]\) | \(4128768\) | \(2.7197\) | \(\Gamma_0(N)\)-optimal |
| 81120.b4 | 81120f2 | \([0, -1, 0, 8817519, -7171379775]\) | \(3834800837445824/3342041015625\) | \(-66074188401000000000000\) | \([2]\) | \(8257536\) | \(3.0663\) | |
| 81120.b2 | 81120f3 | \([0, -1, 0, -19860936, 33374055636]\) | \(350584567631475848/8259273550125\) | \(20411359183465114176000\) | \([2]\) | \(8257536\) | \(3.0663\) | |
| 81120.b1 | 81120f4 | \([0, -1, 0, -38450936, -91730475864]\) | \(2543984126301795848/909361981125\) | \(2247330096513013824000\) | \([2]\) | \(8257536\) | \(3.0663\) |
Rank
sage: E.rank()
The elliptic curves in class 81120f have rank \(1\).
Complex multiplication
The elliptic curves in class 81120f do not have complex multiplication.Modular form 81120.2.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.
