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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 60690p
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
60690.p4 | 60690p1 | \([1, 1, 0, 9098, 4865824]\) | \(3449795831/425079900\) | \(-10260395416763100\) | \([2]\) | \(663552\) | \(1.7516\) | \(\Gamma_0(N)\)-optimal |
60690.p2 | 60690p2 | \([1, 1, 0, -381052, 87499594]\) | \(253503932606569/9180151470\) | \(221586539537576430\) | \([2]\) | \(1327104\) | \(2.0982\) | |
60690.p3 | 60690p3 | \([1, 1, 0, -81937, -131850539]\) | \(-2520453225529/309519000000\) | \(-7471036219311000000\) | \([2]\) | \(1990656\) | \(2.3009\) | |
60690.p1 | 60690p4 | \([1, 1, 0, -4416937, -3546963539]\) | \(394815279796185529/3548222643000\) | \(85645468872774867000\) | \([2]\) | \(3981312\) | \(2.6475\) |
Rank
sage: E.rank()
The elliptic curves in class 60690p have rank \(0\).
Complex multiplication
The elliptic curves in class 60690p do not have complex multiplication.Modular form 60690.2.a.p
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.