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SageMath
E = EllipticCurve("gk1")
E.isogeny_class()
Elliptic curves in class 208080gk
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 208080.ci3 | 208080gk1 | \([0, 0, 0, -2474418, 1228205783]\) | \(5951163357184/1129312125\) | \(317947218698542194000\) | \([2]\) | \(7962624\) | \(2.6516\) | \(\Gamma_0(N)\)-optimal |
| 208080.ci2 | 208080gk2 | \([0, 0, 0, -11955063, -14792188138]\) | \(41948679809104/3291890625\) | \(14828814040809924000000\) | \([2, 2]\) | \(15925248\) | \(2.9982\) | |
| 208080.ci4 | 208080gk3 | \([0, 0, 0, 11922117, -66505384582]\) | \(10400706415004/112060546875\) | \(-2019174025164750000000000\) | \([2]\) | \(31850496\) | \(3.3448\) | |
| 208080.ci1 | 208080gk4 | \([0, 0, 0, -187522563, -988384202638]\) | \(40472803590982276/281883375\) | \(5079143416644821376000\) | \([2]\) | \(31850496\) | \(3.3448\) |
Rank
sage: E.rank()
The elliptic curves in class 208080gk have rank \(0\).
Complex multiplication
The elliptic curves in class 208080gk do not have complex multiplication.Modular form 208080.2.a.gk
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.
