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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 456.d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 456.d1 | 456b3 | \([0, 1, 0, -1272, 16560]\) | \(111223479026/3518667\) | \(7206230016\) | \([2]\) | \(192\) | \(0.66620\) | |
| 456.d2 | 456b2 | \([0, 1, 0, -192, -720]\) | \(768400132/263169\) | \(269485056\) | \([2, 2]\) | \(96\) | \(0.31963\) | |
| 456.d3 | 456b1 | \([0, 1, 0, -172, -928]\) | \(2211014608/513\) | \(131328\) | \([2]\) | \(48\) | \(-0.026945\) | \(\Gamma_0(N)\)-optimal |
| 456.d4 | 456b4 | \([0, 1, 0, 568, -4368]\) | \(9878111854/10097379\) | \(-20679432192\) | \([2]\) | \(192\) | \(0.66620\) |
Rank
sage: E.rank()
The elliptic curves in class 456.d have rank \(0\).
Complex multiplication
The elliptic curves in class 456.d do not have complex multiplication.Modular form 456.2.a.d
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.
