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SageMath
E = EllipticCurve("gv1")
E.isogeny_class()
Elliptic curves in class 405600gv
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
405600.gv3 | 405600gv1 | \([0, 1, 0, -68742158, -124159842312]\) | \(7442744143086784/2927948765625\) | \(14132649453457640625000000\) | \([2, 2]\) | \(99090432\) | \(3.5244\) | \(\Gamma_0(N)\)-optimal* |
405600.gv2 | 405600gv2 | \([0, 1, 0, -496523408, 4170763907688]\) | \(350584567631475848/8259273550125\) | \(318927487241642409000000000\) | \([2]\) | \(198180864\) | \(3.8710\) | \(\Gamma_0(N)\)-optimal* |
405600.gv4 | 405600gv3 | \([0, 1, 0, 220437967, -895981595937]\) | \(3834800837445824/3342041015625\) | \(-1032409193765625000000000000\) | \([2]\) | \(198180864\) | \(3.8710\) | |
405600.gv1 | 405600gv4 | \([0, 1, 0, -961273408, -11468232029812]\) | \(2543984126301795848/909361981125\) | \(35114532758015841000000000\) | \([2]\) | \(198180864\) | \(3.8710\) |
Rank
sage: E.rank()
The elliptic curves in class 405600gv have rank \(1\).
Complex multiplication
The elliptic curves in class 405600gv do not have complex multiplication.Modular form 405600.2.a.gv
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 2 & 2 \\ 2 & 1 & 4 & 4 \\ 2 & 4 & 1 & 4 \\ 2 & 4 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.