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SageMath
E = EllipticCurve("ev1")
E.isogeny_class()
Elliptic curves in class 76050.ev
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
76050.ev1 | 76050ec4 | \([1, -1, 1, -18385880, 30343338497]\) | \(12501706118329/2570490\) | \(141326494534576406250\) | \([2]\) | \(4128768\) | \(2.8635\) | |
76050.ev2 | 76050ec2 | \([1, -1, 1, -1274630, 364428497]\) | \(4165509529/1368900\) | \(75262630225514062500\) | \([2, 2]\) | \(2064384\) | \(2.5170\) | |
76050.ev3 | 76050ec1 | \([1, -1, 1, -514130, -137501503]\) | \(273359449/9360\) | \(514616275046250000\) | \([2]\) | \(1032192\) | \(2.1704\) | \(\Gamma_0(N)\)-optimal |
76050.ev4 | 76050ec3 | \([1, -1, 1, 3668620, 2499912497]\) | \(99317171591/106616250\) | \(-5861801007948691406250\) | \([2]\) | \(4128768\) | \(2.8635\) |
Rank
sage: E.rank()
The elliptic curves in class 76050.ev have rank \(1\).
Complex multiplication
The elliptic curves in class 76050.ev do not have complex multiplication.Modular form 76050.2.a.ev
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.