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SageMath
E = EllipticCurve("gz1")
E.isogeny_class()
Elliptic curves in class 235200.gz
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 235200.gz1 | 235200gz4 | \([0, -1, 0, -785633, -267760863]\) | \(890277128/15\) | \(903544320000000\) | \([2]\) | \(2359296\) | \(2.0000\) | |
| 235200.gz2 | 235200gz3 | \([0, -1, 0, -197633, 29767137]\) | \(14172488/1875\) | \(112943040000000000\) | \([2]\) | \(2359296\) | \(2.0000\) | |
| 235200.gz3 | 235200gz2 | \([0, -1, 0, -50633, -3895863]\) | \(1906624/225\) | \(1694145600000000\) | \([2, 2]\) | \(1179648\) | \(1.6535\) | |
| 235200.gz4 | 235200gz1 | \([0, -1, 0, 4492, -312738]\) | \(85184/405\) | \(-47647845000000\) | \([2]\) | \(589824\) | \(1.3069\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 235200.gz have rank \(0\).
Complex multiplication
The elliptic curves in class 235200.gz do not have complex multiplication.Modular form 235200.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
