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SageMath
E = EllipticCurve("w1")
E.isogeny_class()
Elliptic curves in class 12138w
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
12138.ba5 | 12138w1 | \([1, 0, 0, -20300522, 35159634852]\) | \(38331145780597164097/55468445663232\) | \(1338873434519013163008\) | \([4]\) | \(1105920\) | \(2.9560\) | \(\Gamma_0(N)\)-optimal |
12138.ba4 | 12138w2 | \([1, 0, 0, -26219242, 12984558500]\) | \(82582985847542515777/44772582831427584\) | \(1080701307401798677303296\) | \([2, 2]\) | \(2211840\) | \(3.3026\) | |
12138.ba2 | 12138w3 | \([1, 0, 0, -248263722, -1495363594140]\) | \(70108386184777836280897/552468975892674624\) | \(13335258025968770331349056\) | \([2, 2]\) | \(4423680\) | \(3.6492\) | |
12138.ba6 | 12138w4 | \([1, 0, 0, 101125718, 102151499492]\) | \(4738217997934888496063/2928751705237796928\) | \(-70692946369044984757588032\) | \([2]\) | \(4423680\) | \(3.6492\) | |
12138.ba1 | 12138w5 | \([1, 0, 0, -3964676562, -96086246480388]\) | \(285531136548675601769470657/17941034271597192\) | \(433052952662041962106248\) | \([2]\) | \(8847360\) | \(3.9957\) | |
12138.ba3 | 12138w6 | \([1, 0, 0, -84562562, -3437874298932]\) | \(-2770540998624539614657/209924951154647363208\) | \(-5067077993316930400101161352\) | \([2]\) | \(8847360\) | \(3.9957\) |
Rank
sage: E.rank()
The elliptic curves in class 12138w have rank \(1\).
Complex multiplication
The elliptic curves in class 12138w do not have complex multiplication.Modular form 12138.2.a.w
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.