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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 10002c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
10002.a1 | 10002c1 | \([1, 1, 0, -101, -435]\) | \(115714886617/320064\) | \(320064\) | \([2]\) | \(1536\) | \(-0.070807\) | \(\Gamma_0(N)\)-optimal |
10002.a2 | 10002c2 | \([1, 1, 0, -61, -731]\) | \(-25750777177/200080008\) | \(-200080008\) | \([2]\) | \(3072\) | \(0.27577\) |
Rank
sage: E.rank()
The elliptic curves in class 10002c have rank \(0\).
Complex multiplication
The elliptic curves in class 10002c do not have complex multiplication.Modular form 10002.2.a.c
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.