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SageMath
E = EllipticCurve("o1")
E.isogeny_class()
Elliptic curves in class 51205o
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
51205.d2 | 51205o1 | \([1, 1, 1, -295, 1420]\) | \(24137569/5225\) | \(614716025\) | \([2]\) | \(23040\) | \(0.40034\) | \(\Gamma_0(N)\)-optimal |
51205.d1 | 51205o2 | \([1, 1, 1, -1520, -22100]\) | \(3301293169/218405\) | \(25695129845\) | \([2]\) | \(46080\) | \(0.74691\) |
Rank
sage: E.rank()
The elliptic curves in class 51205o have rank \(0\).
Complex multiplication
The elliptic curves in class 51205o do not have complex multiplication.Modular form 51205.2.a.o
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.