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SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 6045.h
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
6045.h1 | 6045b3 | \([1, 1, 0, -2483, -48672]\) | \(1694053550246329/13280865\) | \(13280865\) | \([2]\) | \(4608\) | \(0.53992\) | |
6045.h2 | 6045b2 | \([1, 1, 0, -158, -777]\) | \(440537367529/36542025\) | \(36542025\) | \([2, 2]\) | \(2304\) | \(0.19335\) | |
6045.h3 | 6045b1 | \([1, 1, 0, -33, 48]\) | \(4165509529/755625\) | \(755625\) | \([2]\) | \(1152\) | \(-0.15323\) | \(\Gamma_0(N)\)-optimal |
6045.h4 | 6045b4 | \([1, 1, 0, 167, -3182]\) | \(510273943271/4862338065\) | \(-4862338065\) | \([2]\) | \(4608\) | \(0.53992\) |
Rank
sage: E.rank()
The elliptic curves in class 6045.h have rank \(0\).
Complex multiplication
The elliptic curves in class 6045.h do not have complex multiplication.Modular form 6045.2.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.