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SageMath
E = EllipticCurve("s1")
E.isogeny_class()
Elliptic curves in class 12675s
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
12675.d1 | 12675s1 | \([0, -1, 1, -1408, -20382]\) | \(-102400/3\) | \(-9050266875\) | \([]\) | \(14040\) | \(0.68938\) | \(\Gamma_0(N)\)-optimal |
12675.d2 | 12675s2 | \([0, -1, 1, 7042, 1002068]\) | \(20480/243\) | \(-458169760546875\) | \([]\) | \(70200\) | \(1.4941\) |
Rank
sage: E.rank()
The elliptic curves in class 12675s have rank \(0\).
Complex multiplication
The elliptic curves in class 12675s do not have complex multiplication.Modular form 12675.2.a.s
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.
\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.