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SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 1575.h
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1575.h1 | 1575f3 | \([1, -1, 0, -25317, 1556716]\) | \(157551496201/13125\) | \(149501953125\) | \([2]\) | \(3072\) | \(1.1881\) | |
| 1575.h2 | 1575f2 | \([1, -1, 0, -1692, 21091]\) | \(47045881/11025\) | \(125581640625\) | \([2, 2]\) | \(1536\) | \(0.84149\) | |
| 1575.h3 | 1575f1 | \([1, -1, 0, -567, -4784]\) | \(1771561/105\) | \(1196015625\) | \([2]\) | \(768\) | \(0.49492\) | \(\Gamma_0(N)\)-optimal |
| 1575.h4 | 1575f4 | \([1, -1, 0, 3933, 127966]\) | \(590589719/972405\) | \(-11076300703125\) | \([2]\) | \(3072\) | \(1.1881\) |
Rank
sage: E.rank()
The elliptic curves in class 1575.h have rank \(0\).
Complex multiplication
The elliptic curves in class 1575.h do not have complex multiplication.Modular form 1575.2.a.h
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
