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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 21675k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
21675.t2 | 21675k1 | \([1, 1, 0, -21825, -87561000]\) | \(-24389/70227\) | \(-3310759879224609375\) | \([2]\) | \(460800\) | \(2.2326\) | \(\Gamma_0(N)\)-optimal |
21675.t1 | 21675k2 | \([1, 1, 0, -3092450, -2068114125]\) | \(69375867029/1003833\) | \(47324391214798828125\) | \([2]\) | \(921600\) | \(2.5792\) |
Rank
sage: E.rank()
The elliptic curves in class 21675k have rank \(0\).
Complex multiplication
The elliptic curves in class 21675k do not have complex multiplication.Modular form 21675.2.a.k
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.