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SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 6672.h
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 6672.h1 | 6672k3 | \([0, -1, 0, -23894312, -44948257680]\) | \(368338718602320108230953/104500400213376\) | \(428033639273988096\) | \([2]\) | \(225792\) | \(2.7510\) | |
| 6672.h2 | 6672k2 | \([0, -1, 0, -1499432, -695974800]\) | \(91021581897882444073/1514074014498816\) | \(6201647163387150336\) | \([2, 2]\) | \(112896\) | \(2.4044\) | |
| 6672.h3 | 6672k1 | \([0, -1, 0, -188712, 14959728]\) | \(181453194188333353/81602499575808\) | \(334243838262509568\) | \([2]\) | \(56448\) | \(2.0579\) | \(\Gamma_0(N)\)-optimal |
| 6672.h4 | 6672k4 | \([0, -1, 0, -76072, -1964473232]\) | \(-11886225803094313/407023891358666112\) | \(-1667169859005096394752\) | \([2]\) | \(225792\) | \(2.7510\) |
Rank
sage: E.rank()
The elliptic curves in class 6672.h have rank \(1\).
Complex multiplication
The elliptic curves in class 6672.h do not have complex multiplication.Modular form 6672.2.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.
