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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 672f
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
672.f3 | 672f1 | \([0, 1, 0, -14, -24]\) | \(5088448/441\) | \(28224\) | \([2, 2]\) | \(64\) | \(-0.40521\) | \(\Gamma_0(N)\)-optimal |
672.f1 | 672f2 | \([0, 1, 0, -224, -1368]\) | \(2438569736/21\) | \(10752\) | \([2]\) | \(128\) | \(-0.058633\) | |
672.f2 | 672f3 | \([0, 1, 0, -49, 95]\) | \(3241792/567\) | \(2322432\) | \([4]\) | \(128\) | \(-0.058633\) | |
672.f4 | 672f4 | \([0, 1, 0, 16, -84]\) | \(830584/7203\) | \(-3687936\) | \([2]\) | \(128\) | \(-0.058633\) |
Rank
sage: E.rank()
The elliptic curves in class 672f have rank \(1\).
Complex multiplication
The elliptic curves in class 672f do not have complex multiplication.Modular form 672.2.a.f
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 2 & 2 \\ 2 & 1 & 4 & 4 \\ 2 & 4 & 1 & 4 \\ 2 & 4 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.