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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 5040.d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
5040.d1 | 5040bd3 | \([0, 0, 0, -16203, -793798]\) | \(157551496201/13125\) | \(39191040000\) | \([2]\) | \(8192\) | \(1.0765\) | |
5040.d2 | 5040bd2 | \([0, 0, 0, -1083, -10582]\) | \(47045881/11025\) | \(32920473600\) | \([2, 2]\) | \(4096\) | \(0.72992\) | |
5040.d3 | 5040bd1 | \([0, 0, 0, -363, 2522]\) | \(1771561/105\) | \(313528320\) | \([2]\) | \(2048\) | \(0.38334\) | \(\Gamma_0(N)\)-optimal |
5040.d4 | 5040bd4 | \([0, 0, 0, 2517, -66022]\) | \(590589719/972405\) | \(-2903585771520\) | \([2]\) | \(8192\) | \(1.0765\) |
Rank
sage: E.rank()
The elliptic curves in class 5040.d have rank \(1\).
Complex multiplication
The elliptic curves in class 5040.d do not have complex multiplication.Modular form 5040.2.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.