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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 118354e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
118354.b4 | 118354e1 | \([1, 0, 1, -10516, -257838]\) | \(3048625/1088\) | \(45892420601408\) | \([2]\) | \(417600\) | \(1.3224\) | \(\Gamma_0(N)\)-optimal |
118354.b3 | 118354e2 | \([1, 0, 1, -149756, -22313454]\) | \(8805624625/2312\) | \(97521393777992\) | \([2]\) | \(835200\) | \(1.6689\) | |
118354.b2 | 118354e3 | \([1, 0, 1, -358616, 82617810]\) | \(120920208625/19652\) | \(828931847112932\) | \([2]\) | \(1252800\) | \(1.8717\) | |
118354.b1 | 118354e4 | \([1, 0, 1, -393426, 65602682]\) | \(159661140625/48275138\) | \(2036271082432917458\) | \([2]\) | \(2505600\) | \(2.2183\) |
Rank
sage: E.rank()
The elliptic curves in class 118354e have rank \(2\).
Complex multiplication
The elliptic curves in class 118354e do not have complex multiplication.Modular form 118354.2.a.e
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.