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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 21175a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
21175.be2 | 21175a1 | \([1, -1, 0, -292, 1491]\) | \(132651/35\) | \(727890625\) | \([2]\) | \(6912\) | \(0.40913\) | \(\Gamma_0(N)\)-optimal |
21175.be1 | 21175a2 | \([1, -1, 0, -1667, -24634]\) | \(24642171/1225\) | \(25476171875\) | \([2]\) | \(13824\) | \(0.75570\) |
Rank
sage: E.rank()
The elliptic curves in class 21175a have rank \(1\).
Complex multiplication
The elliptic curves in class 21175a do not have complex multiplication.Modular form 21175.2.a.a
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.