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SageMath
E = EllipticCurve("ld1")
E.isogeny_class()
Elliptic curves in class 342720ld
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 342720.ld5 | 342720ld1 | \([0, 0, 0, 1762548, -874812944]\) | \(3168685387909439/3563732336640\) | \(-681039855199337840640\) | \([2]\) | \(14155776\) | \(2.6840\) | \(\Gamma_0(N)\)-optimal |
| 342720.ld4 | 342720ld2 | \([0, 0, 0, -10033932, -8240535056]\) | \(584614687782041281/184812061593600\) | \(35318134971232262553600\) | \([2, 2]\) | \(28311552\) | \(3.0305\) | |
| 342720.ld3 | 342720ld3 | \([0, 0, 0, -63302412, 187595704816]\) | \(146796951366228945601/5397929064360000\) | \(1031560308436091535360000\) | \([2, 2]\) | \(56623104\) | \(3.3771\) | |
| 342720.ld2 | 342720ld4 | \([0, 0, 0, -145509132, -675482990096]\) | \(1782900110862842086081/328139630024640\) | \(62708459841247657328640\) | \([2]\) | \(56623104\) | \(3.3771\) | |
| 342720.ld1 | 342720ld5 | \([0, 0, 0, -1003726092, 12239689418224]\) | \(585196747116290735872321/836876053125000\) | \(159929504295321600000000\) | \([2]\) | \(113246208\) | \(3.7237\) | |
| 342720.ld6 | 342720ld6 | \([0, 0, 0, 24825588, 669021343216]\) | \(8854313460877886399/1016927675429790600\) | \(-194337905151395062716825600\) | \([2]\) | \(113246208\) | \(3.7237\) |
Rank
sage: E.rank()
The elliptic curves in class 342720ld have rank \(1\).
Complex multiplication
The elliptic curves in class 342720ld do not have complex multiplication.Modular form 342720.2.a.ld
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
