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SageMath
E = EllipticCurve("pc1")
E.isogeny_class()
Elliptic curves in class 141120.pc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 141120.pc1 | 141120cd3 | \([0, 0, 0, -115676652, 478846840304]\) | \(60910917333827912/3255076125\) | \(9148014683781894144000\) | \([2]\) | \(14155776\) | \(3.2814\) | |
| 141120.pc2 | 141120cd2 | \([0, 0, 0, -7631652, 6603754304]\) | \(139927692143296/27348890625\) | \(9607611443443776000000\) | \([2, 2]\) | \(7077888\) | \(2.9348\) | |
| 141120.pc3 | 141120cd1 | \([0, 0, 0, -2337447, -1282493464]\) | \(257307998572864/19456203375\) | \(106795717943094936000\) | \([2]\) | \(3538944\) | \(2.5882\) | \(\Gamma_0(N)\)-optimal |
| 141120.pc4 | 141120cd4 | \([0, 0, 0, 15706068, 39080525456]\) | \(152461584507448/322998046875\) | \(-907748624664000000000000\) | \([2]\) | \(14155776\) | \(3.2814\) |
Rank
sage: E.rank()
The elliptic curves in class 141120.pc have rank \(1\).
Complex multiplication
The elliptic curves in class 141120.pc do not have complex multiplication.Modular form 141120.2.a.pc
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
