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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 68970.j
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
68970.j1 | 68970o4 | \([1, 1, 0, -59590082, -177080195376]\) | \(13209596798923694545921/92340\) | \(163585942740\) | \([2]\) | \(5529600\) | \(2.6864\) | |
68970.j2 | 68970o3 | \([1, 1, 0, -3770362, -2696245064]\) | \(3345930611358906241/165622259047500\) | \(293409934860448147500\) | \([2]\) | \(5529600\) | \(2.6864\) | |
68970.j3 | 68970o2 | \([1, 1, 0, -3724382, -2768038236]\) | \(3225005357698077121/8526675600\) | \(15105525952611600\) | \([2, 2]\) | \(2764800\) | \(2.3399\) | |
68970.j4 | 68970o1 | \([1, 1, 0, -229902, -44440524]\) | \(-758575480593601/40535043840\) | \(-71810302800234240\) | \([2]\) | \(1382400\) | \(1.9933\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 68970.j have rank \(1\).
Complex multiplication
The elliptic curves in class 68970.j do not have complex multiplication.Modular form 68970.2.a.j
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.