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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 450.b
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 450.b1 | 450c4 | \([1, -1, 0, -7452, -244944]\) | \(502270291349/1889568\) | \(172186884000\) | \([2]\) | \(640\) | \(1.0153\) | |
| 450.b2 | 450c2 | \([1, -1, 0, -477, 4131]\) | \(131872229/18\) | \(1640250\) | \([2]\) | \(128\) | \(0.21061\) | |
| 450.b3 | 450c3 | \([1, -1, 0, -252, -7344]\) | \(-19465109/248832\) | \(-22674816000\) | \([2]\) | \(320\) | \(0.66875\) | |
| 450.b4 | 450c1 | \([1, -1, 0, -27, 81]\) | \(-24389/12\) | \(-1093500\) | \([2]\) | \(64\) | \(-0.13597\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 450.b have rank \(1\).
Complex multiplication
The elliptic curves in class 450.b do not have complex multiplication.Modular form 450.2.a.b
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.
