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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 6864.d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
6864.d1 | 6864c4 | \([0, -1, 0, -1704, 27600]\) | \(267335955794/570999\) | \(1169405952\) | \([2]\) | \(4096\) | \(0.62438\) | |
6864.d2 | 6864c3 | \([0, -1, 0, -1464, -20976]\) | \(169556172914/942513\) | \(1930266624\) | \([2]\) | \(4096\) | \(0.62438\) | |
6864.d3 | 6864c2 | \([0, -1, 0, -144, 144]\) | \(324730948/184041\) | \(188457984\) | \([2, 2]\) | \(2048\) | \(0.27780\) | |
6864.d4 | 6864c1 | \([0, -1, 0, 36, 0]\) | \(19600688/11583\) | \(-2965248\) | \([2]\) | \(1024\) | \(-0.068769\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 6864.d have rank \(1\).
Complex multiplication
The elliptic curves in class 6864.d do not have complex multiplication.Modular form 6864.2.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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.