Show commands:
SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 60333d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 60333.l4 | 60333d1 | \([1, 1, 0, -277839, 56253072]\) | \(491411892194497/78897\) | \(380820749673\) | \([2]\) | \(258048\) | \(1.6246\) | \(\Gamma_0(N)\)-optimal |
| 60333.l3 | 60333d2 | \([1, 1, 0, -278684, 55892595]\) | \(495909170514577/6224736609\) | \(30045614686950681\) | \([2, 2]\) | \(516096\) | \(1.9712\) | |
| 60333.l5 | 60333d3 | \([1, 1, 0, -47999, 145905882]\) | \(-2533811507137/1904381781393\) | \(-9192087121863764937\) | \([2]\) | \(1032192\) | \(2.3178\) | |
| 60333.l2 | 60333d4 | \([1, 1, 0, -522889, -57174320]\) | \(3275619238041697/1605271262049\) | \(7748337775099471641\) | \([2, 2]\) | \(1032192\) | \(2.3178\) | |
| 60333.l6 | 60333d5 | \([1, 1, 0, 1904796, -435407643]\) | \(158346567380527343/108665074944153\) | \(-524505561726112197777\) | \([2]\) | \(2064384\) | \(2.6644\) | |
| 60333.l1 | 60333d6 | \([1, 1, 0, -6857854, -6910339457]\) | \(7389727131216686257/6115533215337\) | \(29518510763587569633\) | \([2]\) | \(2064384\) | \(2.6644\) |
Rank
sage: E.rank()
The elliptic curves in class 60333d have rank \(1\).
Complex multiplication
The elliptic curves in class 60333d do not have complex multiplication.Modular form 60333.2.a.d
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
