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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 68400.i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
68400.i1 | 68400ft4 | \([0, 0, 0, -3483075, -2501522750]\) | \(100162392144121/23457780\) | \(1094446183680000000\) | \([2]\) | \(2359296\) | \(2.4519\) | |
68400.i2 | 68400ft3 | \([0, 0, 0, -1611075, 765405250]\) | \(9912050027641/311647500\) | \(14540225760000000000\) | \([2]\) | \(2359296\) | \(2.4519\) | |
68400.i3 | 68400ft2 | \([0, 0, 0, -243075, -29402750]\) | \(34043726521/11696400\) | \(545707238400000000\) | \([2, 2]\) | \(1179648\) | \(2.1053\) | |
68400.i4 | 68400ft1 | \([0, 0, 0, 44925, -3194750]\) | \(214921799/218880\) | \(-10212065280000000\) | \([2]\) | \(589824\) | \(1.7587\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 68400.i have rank \(2\).
Complex multiplication
The elliptic curves in class 68400.i do not have complex multiplication.Modular form 68400.2.a.i
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.