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SageMath
E = EllipticCurve("g1")
E.isogeny_class()
Elliptic curves in class 1568g
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality | CM discriminant |
---|---|---|---|---|---|---|---|---|---|
1568.e3 | 1568g1 | \([0, 0, 0, -49, 0]\) | \(1728\) | \(7529536\) | \([2, 2]\) | \(192\) | \(0.0089957\) | \(\Gamma_0(N)\)-optimal | \(-4\) |
1568.e1 | 1568g2 | \([0, 0, 0, -539, -4802]\) | \(287496\) | \(60236288\) | \([2]\) | \(384\) | \(0.35557\) | \(-16\) | |
1568.e2 | 1568g3 | \([0, 0, 0, -539, 4802]\) | \(287496\) | \(60236288\) | \([2]\) | \(384\) | \(0.35557\) | \(-16\) | |
1568.e4 | 1568g4 | \([0, 0, 0, 196, 0]\) | \(1728\) | \(-481890304\) | \([2]\) | \(384\) | \(0.35557\) | \(-4\) |
Rank
sage: E.rank()
The elliptic curves in class 1568g have rank \(1\).
Complex multiplication
Each elliptic curve in class 1568g has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-1}) \).Modular form 1568.2.a.g
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.