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SageMath
E = EllipticCurve("cm1")
E.isogeny_class()
Elliptic curves in class 185150cm
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
185150.z3 | 185150cm1 | \([1, 1, 0, -1997250, -1055687500]\) | \(380920459249/12622400\) | \(29196378208025000000\) | \([2]\) | \(7299072\) | \(2.5081\) | \(\Gamma_0(N)\)-optimal |
185150.z4 | 185150cm2 | \([1, 1, 0, 647750, -3645142500]\) | \(12994449551/2489452840\) | \(-5758255692077730625000\) | \([2]\) | \(14598144\) | \(2.8547\) | |
185150.z1 | 185150cm3 | \([1, 1, 0, -22363750, 40345704000]\) | \(534774372149809/5323062500\) | \(12312567021719726562500\) | \([2]\) | \(21897216\) | \(3.0574\) | |
185150.z2 | 185150cm4 | \([1, 1, 0, -5832500, 98684485250]\) | \(-9486391169809/1813439640250\) | \(-4194596082691389566406250\) | \([2]\) | \(43794432\) | \(3.4040\) |
Rank
sage: E.rank()
The elliptic curves in class 185150cm have rank \(1\).
Complex multiplication
The elliptic curves in class 185150cm do not have complex multiplication.Modular form 185150.2.a.cm
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.