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SageMath
E = EllipticCurve("kd1")
E.isogeny_class()
Elliptic curves in class 352800kd
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
352800.kd1 | 352800kd1 | \([0, 0, 0, -334425, 67228000]\) | \(140608/15\) | \(441266692545000000\) | \([2]\) | \(3440640\) | \(2.1201\) | \(\Gamma_0(N)\)-optimal |
352800.kd2 | 352800kd2 | \([0, 0, 0, 437325, 331938250]\) | \(39304/225\) | \(-52952003105400000000\) | \([2]\) | \(6881280\) | \(2.4666\) |
Rank
sage: E.rank()
The elliptic curves in class 352800kd have rank \(1\).
Complex multiplication
The elliptic curves in class 352800kd do not have complex multiplication.Modular form 352800.2.a.kd
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.