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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 392784.f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 392784.f1 | 392784f3 | \([0, -1, 0, -17518516864, 892476847550464]\) | \(1233864675106127856683588593/27488595456\) | \(13246487620824858624\) | \([2]\) | \(222953472\) | \(4.1222\) | |
| 392784.f2 | 392784f4 | \([0, -1, 0, -1112971904, 13461281107968]\) | \(316393918884564908858353/20661539369919533568\) | \(9956595488078492486621724672\) | \([2]\) | \(222953472\) | \(4.1222\) | |
| 392784.f3 | 392784f2 | \([0, -1, 0, -1094908544, 13945191297024]\) | \(301237516670332318563313/1421837758365696\) | \(685169829617523788611584\) | \([2, 2]\) | \(111476736\) | \(3.7756\) | |
| 392784.f4 | 392784f1 | \([0, -1, 0, -67304064, 225438363648]\) | \(-69967989877865233393/5060983303176192\) | \(-2438838782506499328442368\) | \([2]\) | \(55738368\) | \(3.4291\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 392784.f have rank \(1\).
Complex multiplication
The elliptic curves in class 392784.f do not have complex multiplication.Modular form 392784.2.a.f
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
