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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 42483.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
42483.e1 | 42483t1 | \([1, 0, 0, -1128, -10305]\) | \(274625/81\) | \(46818772497\) | \([2]\) | \(27648\) | \(0.75322\) | \(\Gamma_0(N)\)-optimal |
42483.e2 | 42483t2 | \([1, 0, 0, 3037, -67782]\) | \(5359375/6561\) | \(-3792320572257\) | \([2]\) | \(55296\) | \(1.0998\) |
Rank
sage: E.rank()
The elliptic curves in class 42483.e have rank \(1\).
Complex multiplication
The elliptic curves in class 42483.e do not have complex multiplication.Modular form 42483.2.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 LMFDB numbering.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.