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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 1710.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1710.e1 | 1710g3 | \([1, -1, 0, -4432320, -3590548484]\) | \(13209596798923694545921/92340\) | \(67315860\) | \([2]\) | \(30720\) | \(2.0368\) | |
1710.e2 | 1710g4 | \([1, -1, 0, -280440, -54592700]\) | \(3345930611358906241/165622259047500\) | \(120738626845627500\) | \([2]\) | \(30720\) | \(2.0368\) | |
1710.e3 | 1710g2 | \([1, -1, 0, -277020, -56050304]\) | \(3225005357698077121/8526675600\) | \(6215946512400\) | \([2, 2]\) | \(15360\) | \(1.6902\) | |
1710.e4 | 1710g1 | \([1, -1, 0, -17100, -895280]\) | \(-758575480593601/40535043840\) | \(-29550046959360\) | \([2]\) | \(7680\) | \(1.3436\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 1710.e have rank \(1\).
Complex multiplication
The elliptic curves in class 1710.e do not have complex multiplication.Modular form 1710.2.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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.