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SageMath
E = EllipticCurve("mc1")
E.isogeny_class()
Elliptic curves in class 366912mc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
366912.mc4 | 366912mc1 | \([0, 0, 0, -40719, -451388]\) | \(1360251712/771147\) | \(4232850362290368\) | \([2]\) | \(2359296\) | \(1.6884\) | \(\Gamma_0(N)\)-optimal |
366912.mc2 | 366912mc2 | \([0, 0, 0, -413364, 101802400]\) | \(22235451328/123201\) | \(43280268793122816\) | \([2, 2]\) | \(4718592\) | \(2.0350\) | |
366912.mc1 | 366912mc3 | \([0, 0, 0, -6605004, 6533678032]\) | \(11339065490696/351\) | \(986444872777728\) | \([2]\) | \(9437184\) | \(2.3815\) | |
366912.mc3 | 366912mc4 | \([0, 0, 0, -184044, 214169200]\) | \(-245314376/6908733\) | \(-19416194430884020224\) | \([2]\) | \(9437184\) | \(2.3815\) |
Rank
sage: E.rank()
The elliptic curves in class 366912mc have rank \(0\).
Complex multiplication
The elliptic curves in class 366912mc do not have complex multiplication.Modular form 366912.2.a.mc
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.