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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 13680.p
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 13680.p1 | 13680m5 | \([0, 0, 0, -6238083, -5996872798]\) | \(17981241677724245762/16245\) | \(24253655040\) | \([2]\) | \(131072\) | \(2.1879\) | |
| 13680.p2 | 13680m3 | \([0, 0, 0, -389883, -93699718]\) | \(8780093172522724/263900025\) | \(197000313062400\) | \([2, 2]\) | \(65536\) | \(1.8413\) | |
| 13680.p3 | 13680m6 | \([0, 0, 0, -373683, -101841838]\) | \(-3865238121540962/764260336845\) | \(-1141034568826890240\) | \([2]\) | \(131072\) | \(2.1879\) | |
| 13680.p4 | 13680m4 | \([0, 0, 0, -110883, 12908882]\) | \(201971983086724/20447192475\) | \(15263747393817600\) | \([2]\) | \(65536\) | \(1.8413\) | |
| 13680.p5 | 13680m2 | \([0, 0, 0, -25383, -1335418]\) | \(9691367618896/1480325625\) | \(276264289440000\) | \([2, 2]\) | \(32768\) | \(1.4948\) | |
| 13680.p6 | 13680m1 | \([0, 0, 0, 2742, -114793]\) | \(195469297664/601171875\) | \(-7012068750000\) | \([2]\) | \(16384\) | \(1.1482\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 13680.p have rank \(1\).
Complex multiplication
The elliptic curves in class 13680.p do not have complex multiplication.Modular form 13680.2.a.p
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
