Show commands:
SageMath
E = EllipticCurve("jf1")
E.isogeny_class()
Elliptic curves in class 374850.jf
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
374850.jf1 | 374850jf6 | \([1, -1, 1, -19211944730, -1024949657573103]\) | \(585196747116290735872321/836876053125000\) | \(1121493950536268408203125000\) | \([2]\) | \(679477248\) | \(4.4616\) | |
374850.jf2 | 374850jf3 | \([1, -1, 1, -2785135730, 56565800190897]\) | \(1782900110862842086081/328139630024640\) | \(439738487712320425335000000\) | \([2]\) | \(339738624\) | \(4.1151\) | |
374850.jf3 | 374850jf4 | \([1, -1, 1, -1211647730, -15709005377103]\) | \(146796951366228945601/5397929064360000\) | \(7233741207551821055625000000\) | \([2, 2]\) | \(339738624\) | \(4.1151\) | |
374850.jf4 | 374850jf2 | \([1, -1, 1, -192055730, 690112350897]\) | \(584614687782041281/184812061593600\) | \(247665838076502350400000000\) | \([2, 2]\) | \(169869312\) | \(3.7685\) | |
374850.jf5 | 374850jf1 | \([1, -1, 1, 33736270, 73248606897]\) | \(3168685387909439/3563732336640\) | \(-4775742168685608960000000\) | \([2]\) | \(84934656\) | \(3.4219\) | \(\Gamma_0(N)\)-optimal |
374850.jf6 | 374850jf5 | \([1, -1, 1, 475177270, -56024122877103]\) | \(8854313460877886399/1016927675429790600\) | \(-1362780344674377306316603125000\) | \([2]\) | \(679477248\) | \(4.4616\) |
Rank
sage: E.rank()
The elliptic curves in class 374850.jf have rank \(0\).
Complex multiplication
The elliptic curves in class 374850.jf do not have complex multiplication.Modular form 374850.2.a.jf
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 8 & 4 \\ 8 & 1 & 4 & 2 & 4 & 8 \\ 2 & 4 & 1 & 2 & 4 & 2 \\ 4 & 2 & 2 & 1 & 2 & 4 \\ 8 & 4 & 4 & 2 & 1 & 8 \\ 4 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.