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SageMath
E = EllipticCurve("cr1")
E.isogeny_class()
Elliptic curves in class 279174.cr
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 279174.cr1 | 279174cr6 | \([1, 0, 0, -22869732, -42097375692]\) | \(54804145548726848737/637608031452\) | \(15390307854126820188\) | \([2]\) | \(18874368\) | \(2.8326\) | |
| 279174.cr2 | 279174cr3 | \([1, 0, 0, -5119352, 4457767248]\) | \(614716917569296417/19093020912\) | \(460859109681842928\) | \([2]\) | \(9437184\) | \(2.4860\) | |
| 279174.cr3 | 279174cr4 | \([1, 0, 0, -1466392, -621983440]\) | \(14447092394873377/1439452851984\) | \(34744892537010586896\) | \([2, 2]\) | \(9437184\) | \(2.4860\) | |
| 279174.cr4 | 279174cr2 | \([1, 0, 0, -333512, 63408960]\) | \(169967019783457/26337394944\) | \(635720687741051136\) | \([2, 2]\) | \(4718592\) | \(2.1394\) | |
| 279174.cr5 | 279174cr1 | \([1, 0, 0, 36408, 5479488]\) | \(221115865823/664731648\) | \(-16045006020083712\) | \([2]\) | \(2359296\) | \(1.7929\) | \(\Gamma_0(N)\)-optimal |
| 279174.cr6 | 279174cr5 | \([1, 0, 0, 1810868, -3007173268]\) | \(27207619911317663/177609314617308\) | \(-4287057086617980444252\) | \([2]\) | \(18874368\) | \(2.8326\) |
Rank
sage: E.rank()
The elliptic curves in class 279174.cr have rank \(0\).
Complex multiplication
The elliptic curves in class 279174.cr do not have complex multiplication.Modular form 279174.2.a.cr
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.
