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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 21175.p
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
21175.p1 | 21175bl2 | \([1, 1, 1, -157968, 24099956]\) | \(1968634623437/5929\) | \(1312948146125\) | \([2]\) | \(99840\) | \(1.5531\) | |
21175.p2 | 21175bl1 | \([1, 1, 1, -9743, 383956]\) | \(-461889917/26411\) | \(-5848587196375\) | \([2]\) | \(49920\) | \(1.2065\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 21175.p have rank \(0\).
Complex multiplication
The elliptic curves in class 21175.p do not have complex multiplication.Modular form 21175.2.a.p
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.