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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 450.f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 450.f1 | 450a4 | \([1, -1, 1, -186305, -30804303]\) | \(502270291349/1889568\) | \(2690420062500000\) | \([2]\) | \(3200\) | \(1.8200\) | |
| 450.f2 | 450a2 | \([1, -1, 1, -11930, 504447]\) | \(131872229/18\) | \(25628906250\) | \([2]\) | \(640\) | \(1.0153\) | |
| 450.f3 | 450a3 | \([1, -1, 1, -6305, -924303]\) | \(-19465109/248832\) | \(-354294000000000\) | \([2]\) | \(1600\) | \(1.4735\) | |
| 450.f4 | 450a1 | \([1, -1, 1, -680, 9447]\) | \(-24389/12\) | \(-17085937500\) | \([2]\) | \(320\) | \(0.66875\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 450.f have rank \(0\).
Complex multiplication
The elliptic curves in class 450.f do not have complex multiplication.Modular form 450.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 5 & 2 & 10 \\ 5 & 1 & 10 & 2 \\ 2 & 10 & 1 & 5 \\ 10 & 2 & 5 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
