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SageMath
E = EllipticCurve("g1")
E.isogeny_class()
Elliptic curves in class 450.g
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
450.g1 | 450b2 | \([1, -1, 1, -1130, 14897]\) | \(-349938025/8\) | \(-3645000\) | \([3]\) | \(180\) | \(0.37251\) | |
450.g2 | 450b3 | \([1, -1, 1, -680, -8053]\) | \(-121945/32\) | \(-9112500000\) | \([]\) | \(300\) | \(0.62793\) | |
450.g3 | 450b1 | \([1, -1, 1, -5, 47]\) | \(-25/2\) | \(-911250\) | \([]\) | \(60\) | \(-0.17679\) | \(\Gamma_0(N)\)-optimal |
450.g4 | 450b4 | \([1, -1, 1, 4945, 59447]\) | \(46969655/32768\) | \(-9331200000000\) | \([3]\) | \(900\) | \(1.1772\) |
Rank
sage: E.rank()
The elliptic curves in class 450.g have rank \(0\).
Complex multiplication
The elliptic curves in class 450.g do not have complex multiplication.Modular form 450.2.a.g
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.
\(\left(\begin{array}{rrrr} 1 & 15 & 3 & 5 \\ 15 & 1 & 5 & 3 \\ 3 & 5 & 1 & 15 \\ 5 & 3 & 15 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.