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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 99d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
99.d3 | 99d1 | \([0, 0, 1, -3, -5]\) | \(-4096/11\) | \(-8019\) | \([]\) | \(6\) | \(-0.56342\) | \(\Gamma_0(N)\)-optimal |
99.d2 | 99d2 | \([0, 0, 1, -93, 625]\) | \(-122023936/161051\) | \(-117406179\) | \([]\) | \(30\) | \(0.24130\) | |
99.d1 | 99d3 | \([0, 0, 1, -70383, 7187035]\) | \(-52893159101157376/11\) | \(-8019\) | \([]\) | \(150\) | \(1.0460\) |
Rank
sage: E.rank()
The elliptic curves in class 99d have rank \(0\).
Complex multiplication
The elliptic curves in class 99d do not have complex multiplication.Modular form 99.2.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 Cremona numbering.
\(\left(\begin{array}{rrr} 1 & 5 & 25 \\ 5 & 1 & 5 \\ 25 & 5 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.