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SageMath
E = EllipticCurve("es1")
E.isogeny_class()
Elliptic curves in class 126126.es
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 126126.es1 | 126126fl4 | \([1, -1, 1, -28742405, -59297317635]\) | \(30618029936661765625/3678951124992\) | \(315529367339149996032\) | \([2]\) | \(7962624\) | \(2.9581\) | |
| 126126.es2 | 126126fl3 | \([1, -1, 1, -1647365, -1086333699]\) | \(-5764706497797625/2612665516032\) | \(-224078186780527951872\) | \([2]\) | \(3981312\) | \(2.6115\) | |
| 126126.es3 | 126126fl2 | \([1, -1, 1, -794030, 155360013]\) | \(645532578015625/252306960048\) | \(21639389264618933808\) | \([2]\) | \(2654208\) | \(2.4088\) | |
| 126126.es4 | 126126fl1 | \([1, -1, 1, 158530, 17429325]\) | \(5137417856375/4510142208\) | \(-386817402338535168\) | \([2]\) | \(1327104\) | \(2.0622\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 126126.es have rank \(1\).
Complex multiplication
The elliptic curves in class 126126.es do not have complex multiplication.Modular form 126126.2.a.es
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
