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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 139230p
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 139230.dv3 | 139230p1 | \([1, -1, 1, -639167, 196806759]\) | \(39613077168432499369/8661219840000\) | \(6314029263360000\) | \([4]\) | \(1376256\) | \(2.0257\) | \(\Gamma_0(N)\)-optimal |
| 139230.dv2 | 139230p2 | \([1, -1, 1, -711167, 149776359]\) | \(54564527576482291369/18314631132033600\) | \(13351366095252494400\) | \([2, 2]\) | \(2752512\) | \(2.3723\) | |
| 139230.dv4 | 139230p3 | \([1, -1, 1, 2073433, 1033051479]\) | \(1352279296967264534231/1415615917112986680\) | \(-1031984003575367289720\) | \([2]\) | \(5505024\) | \(2.7189\) | |
| 139230.dv1 | 139230p4 | \([1, -1, 1, -4647767, -3744308361]\) | \(15231025329261085948969/501037266310733880\) | \(365256167140524998520\) | \([2]\) | \(5505024\) | \(2.7189\) |
Rank
sage: E.rank()
The elliptic curves in class 139230p have rank \(0\).
Complex multiplication
The elliptic curves in class 139230p do not have complex multiplication.Modular form 139230.2.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 Cremona 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 Cremona labels.
