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SageMath
E = EllipticCurve("gz1")
E.isogeny_class()
Elliptic curves in class 54450gz
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 54450.gp4 | 54450gz1 | \([1, -1, 1, -3290, -97963]\) | \(-24389/12\) | \(-1937201953500\) | \([2]\) | \(89600\) | \(1.0630\) | \(\Gamma_0(N)\)-optimal |
| 54450.gp2 | 54450gz2 | \([1, -1, 1, -57740, -5325163]\) | \(131872229/18\) | \(2905802930250\) | \([2]\) | \(179200\) | \(1.4096\) | |
| 54450.gp3 | 54450gz3 | \([1, -1, 1, -30515, 9866387]\) | \(-19465109/248832\) | \(-40169819707776000\) | \([2]\) | \(448000\) | \(1.8677\) | |
| 54450.gp1 | 54450gz4 | \([1, -1, 1, -901715, 328725587]\) | \(502270291349/1889568\) | \(305039568405924000\) | \([2]\) | \(896000\) | \(2.2143\) |
Rank
sage: E.rank()
The elliptic curves in class 54450gz have rank \(1\).
Complex multiplication
The elliptic curves in class 54450gz do not have complex multiplication.Modular form 54450.2.a.gz
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
