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SageMath
E = EllipticCurve("w1")
E.isogeny_class()
Elliptic curves in class 171462w
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
171462.b2 | 171462w1 | \([1, 1, 0, -429530, 108133332]\) | \(1845026709625/793152\) | \(3767554678957632\) | \([2]\) | \(1658880\) | \(1.9489\) | \(\Gamma_0(N)\)-optimal |
171462.b3 | 171462w2 | \([1, 1, 0, -362290, 143219164]\) | \(-1107111813625/1228691592\) | \(-5836413142040241672\) | \([2]\) | \(3317760\) | \(2.2955\) | |
171462.b1 | 171462w3 | \([1, 1, 0, -1261625, -412894299]\) | \(46753267515625/11591221248\) | \(55059509208494112768\) | \([2]\) | \(4976640\) | \(2.4982\) | |
171462.b4 | 171462w4 | \([1, 1, 0, 3041735, -2613632603]\) | \(655215969476375/1001033261568\) | \(-4755012341156219109888\) | \([2]\) | \(9953280\) | \(2.8448\) |
Rank
sage: E.rank()
The elliptic curves in class 171462w have rank \(2\).
Complex multiplication
The elliptic curves in class 171462w do not have complex multiplication.Modular form 171462.2.a.w
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.