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SageMath
E = EllipticCurve("ck1")
E.isogeny_class()
Elliptic curves in class 130536.ck
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 130536.ck1 | 130536cd3 | \([0, 0, 0, -51170259, 140888172670]\) | \(168717772351634692/5439\) | \(477677498489856\) | \([2]\) | \(5505024\) | \(2.7685\) | |
| 130536.ck2 | 130536cd4 | \([0, 0, 0, -3524619, 1724656822]\) | \(55137176303332/17277108597\) | \(1517353560535492850688\) | \([2]\) | \(5505024\) | \(2.7685\) | |
| 130536.ck3 | 130536cd2 | \([0, 0, 0, -3198279, 2201178490]\) | \(164784750161488/29582721\) | \(649521978571581696\) | \([2, 2]\) | \(2752512\) | \(2.4219\) | |
| 130536.ck4 | 130536cd1 | \([0, 0, 0, -179634, 41639857]\) | \(-467147020288/275501667\) | \(-378059348921979312\) | \([2]\) | \(1376256\) | \(2.0753\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 130536.ck have rank \(0\).
Complex multiplication
The elliptic curves in class 130536.ck do not have complex multiplication.Modular form 130536.2.a.ck
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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
