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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 143143k
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 143143.k4 | 143143k1 | \([1, -1, 1, -320794, 589764816]\) | \(-426957777/17320303\) | \(-148105678494598071247\) | \([2]\) | \(3064320\) | \(2.5505\) | \(\Gamma_0(N)\)-optimal |
| 143143.k3 | 143143k2 | \([1, -1, 1, -12692439, 17311280198]\) | \(26444947540257/169338169\) | \(1448008410405863787481\) | \([2, 2]\) | \(6128640\) | \(2.8971\) | |
| 143143.k1 | 143143k3 | \([1, -1, 1, -202765894, 1111374087178]\) | \(107818231938348177/4463459\) | \(38166978004242778691\) | \([2]\) | \(12257280\) | \(3.2436\) | |
| 143143.k2 | 143143k4 | \([1, -1, 1, -20565304, -6723002074]\) | \(112489728522417/62811265517\) | \(537098288436387995249933\) | \([2]\) | \(12257280\) | \(3.2436\) |
Rank
sage: E.rank()
The elliptic curves in class 143143k have rank \(0\).
Complex multiplication
The elliptic curves in class 143143k do not have complex multiplication.Modular form 143143.2.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.
