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SageMath
E = EllipticCurve("l1")
E.isogeny_class()
Elliptic curves in class 41405l
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
41405.l3 | 41405l1 | \([1, -1, 0, -552239, -157805712]\) | \(32798729601/3185\) | \(1808663567750585\) | \([2]\) | \(258048\) | \(1.9634\) | \(\Gamma_0(N)\)-optimal |
41405.l2 | 41405l2 | \([1, -1, 0, -593644, -132739125]\) | \(40743095121/10144225\) | \(5760593463285613225\) | \([2, 2]\) | \(516096\) | \(2.3100\) | |
41405.l4 | 41405l3 | \([1, -1, 0, 1435201, -844052182]\) | \(575722725759/874680625\) | \(-496704232293504405625\) | \([2]\) | \(1032192\) | \(2.6566\) | |
41405.l1 | 41405l4 | \([1, -1, 0, -3284969, 2182338640]\) | \(6903498885921/374712065\) | \(212787460082288574665\) | \([2]\) | \(1032192\) | \(2.6566\) |
Rank
sage: E.rank()
The elliptic curves in class 41405l have rank \(1\).
Complex multiplication
The elliptic curves in class 41405l do not have complex multiplication.Modular form 41405.2.a.l
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.