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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 2205k
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 2205.b3 | 2205k1 | \([1, -1, 1, -1112, 13794]\) | \(1771561/105\) | \(9005442705\) | \([4]\) | \(1536\) | \(0.66315\) | \(\Gamma_0(N)\)-optimal |
| 2205.b2 | 2205k2 | \([1, -1, 1, -3317, -55884]\) | \(47045881/11025\) | \(945571484025\) | \([2, 2]\) | \(3072\) | \(1.0097\) | |
| 2205.b1 | 2205k3 | \([1, -1, 1, -49622, -4241856]\) | \(157551496201/13125\) | \(1125680338125\) | \([2]\) | \(6144\) | \(1.3563\) | |
| 2205.b4 | 2205k4 | \([1, -1, 1, 7708, -355764]\) | \(590589719/972405\) | \(-83399404891005\) | \([2]\) | \(6144\) | \(1.3563\) |
Rank
sage: E.rank()
The elliptic curves in class 2205k have rank \(0\).
Complex multiplication
The elliptic curves in class 2205k do not have complex multiplication.Modular form 2205.2.a.k
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.
