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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 13680.j
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
13680.j1 | 13680bc4 | \([0, 0, 0, -1076763, -415287862]\) | \(46237740924063961/1806561830400\) | \(5394364720585113600\) | \([2]\) | \(165888\) | \(2.3616\) | |
13680.j2 | 13680bc2 | \([0, 0, 0, -158763, 24174938]\) | \(148212258825961/1218375000\) | \(3638048256000000\) | \([2]\) | \(55296\) | \(1.8122\) | |
13680.j3 | 13680bc1 | \([0, 0, 0, -3243, 878042]\) | \(-1263214441/110808000\) | \(-330870915072000\) | \([2]\) | \(27648\) | \(1.4657\) | \(\Gamma_0(N)\)-optimal |
13680.j4 | 13680bc3 | \([0, 0, 0, 29157, -23570998]\) | \(918046641959/80912056320\) | \(-241602105578618880\) | \([2]\) | \(82944\) | \(2.0150\) |
Rank
sage: E.rank()
The elliptic curves in class 13680.j have rank \(1\).
Complex multiplication
The elliptic curves in class 13680.j do not have complex multiplication.Modular form 13680.2.a.j
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.