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SageMath
E = EllipticCurve("cb1")
E.isogeny_class()
Elliptic curves in class 248430.cb
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 248430.cb1 | 248430cb7 | \([1, 1, 0, -332817158027, 73901943591424551]\) | \(7179471593960193209684686321/49441793310\) | \(28076474186515417645710\) | \([2]\) | \(891813888\) | \(4.8432\) | |
| 248430.cb2 | 248430cb6 | \([1, 1, 0, -20801085477, 1154709841032441]\) | \(1752803993935029634719121/4599740941532100\) | \(2612051448050198739592016100\) | \([2, 2]\) | \(445906944\) | \(4.4966\) | |
| 248430.cb3 | 248430cb8 | \([1, 1, 0, -20545451007, 1184474231531739]\) | \(-1688971789881664420008241/89901485966373558750\) | \(-51052289593099010884561870908750\) | \([2]\) | \(891813888\) | \(4.8432\) | |
| 248430.cb4 | 248430cb4 | \([1, 1, 0, -4110729977, 101275673419341]\) | \(13527956825588849127121/25701087819771000\) | \(14594857516853413181902611000\) | \([2]\) | \(297271296\) | \(4.2939\) | |
| 248430.cb5 | 248430cb3 | \([1, 1, 0, -1316058097, 17575334152069]\) | \(443915739051786565201/21894701746029840\) | \(12433327904178741982066903440\) | \([2]\) | \(222953472\) | \(4.1501\) | |
| 248430.cb6 | 248430cb2 | \([1, 1, 0, -342874977, 432048628341]\) | \(7850236389974007121/4400862921000000\) | \(2499114735283240471761000000\) | \([2, 2]\) | \(148635648\) | \(3.9473\) | |
| 248430.cb7 | 248430cb1 | \([1, 1, 0, -213028897, -1190430079691]\) | \(1882742462388824401/11650189824000\) | \(6615784581490549430784000\) | \([2]\) | \(74317824\) | \(3.6008\) | \(\Gamma_0(N)\)-optimal |
| 248430.cb8 | 248430cb5 | \([1, 1, 0, 1347442743, 3429658072989]\) | \(476437916651992691759/284661685546875000\) | \(-161650618456234246294921875000\) | \([2]\) | \(297271296\) | \(4.2939\) |
Rank
sage: E.rank()
The elliptic curves in class 248430.cb have rank \(1\).
Complex multiplication
The elliptic curves in class 248430.cb do not have complex multiplication.Modular form 248430.2.a.cb
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 3 & 4 & 6 & 12 & 12 \\ 2 & 1 & 2 & 6 & 2 & 3 & 6 & 6 \\ 4 & 2 & 1 & 12 & 4 & 6 & 12 & 3 \\ 3 & 6 & 12 & 1 & 12 & 2 & 4 & 4 \\ 4 & 2 & 4 & 12 & 1 & 6 & 3 & 12 \\ 6 & 3 & 6 & 2 & 6 & 1 & 2 & 2 \\ 12 & 6 & 12 & 4 & 3 & 2 & 1 & 4 \\ 12 & 6 & 3 & 4 & 12 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
