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SageMath
E = EllipticCurve("o1")
E.isogeny_class()
Elliptic curves in class 12870.o
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 12870.o1 | 12870bc4 | \([1, -1, 0, -31943899899, 2197513725059205]\) | \(4944928228995290413834018379264689/189679641808585500000\) | \(138276458878458829500000\) | \([6]\) | \(19353600\) | \(4.3804\) | |
| 12870.o2 | 12870bc3 | \([1, -1, 0, -1996399899, 34339915559205]\) | \(-1207087636168285491836819264689/236446260657750000000000\) | \(-172369324019499750000000000\) | \([6]\) | \(9676800\) | \(4.0339\) | |
| 12870.o3 | 12870bc2 | \([1, -1, 0, -397562139, 2963216174373]\) | \(9532597152396244075685450929/313550122650789880627200\) | \(228578039412425822977228800\) | \([2]\) | \(6451200\) | \(3.8311\) | |
| 12870.o4 | 12870bc1 | \([1, -1, 0, 7941861, 159642619173]\) | \(75991146714893572533071/15147028085515223040000\) | \(-11042183474340597596160000\) | \([2]\) | \(3225600\) | \(3.4845\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 12870.o have rank \(1\).
Complex multiplication
The elliptic curves in class 12870.o do not have complex multiplication.Modular form 12870.2.a.o
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.
