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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 720j
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
720.j8 | 720j1 | \([0, 0, 0, 213, 3674]\) | \(357911/2160\) | \(-6449725440\) | \([2]\) | \(384\) | \(0.56417\) | \(\Gamma_0(N)\)-optimal |
720.j6 | 720j2 | \([0, 0, 0, -2667, 48026]\) | \(702595369/72900\) | \(217678233600\) | \([2, 2]\) | \(768\) | \(0.91074\) | |
720.j7 | 720j3 | \([0, 0, 0, -1947, -108214]\) | \(-273359449/1536000\) | \(-4586471424000\) | \([2]\) | \(1152\) | \(1.1135\) | |
720.j5 | 720j4 | \([0, 0, 0, -9867, -324934]\) | \(35578826569/5314410\) | \(15868743229440\) | \([2]\) | \(1536\) | \(1.2573\) | |
720.j4 | 720j5 | \([0, 0, 0, -41547, 3259514]\) | \(2656166199049/33750\) | \(100776960000\) | \([4]\) | \(1536\) | \(1.2573\) | |
720.j3 | 720j6 | \([0, 0, 0, -48027, -4043446]\) | \(4102915888729/9000000\) | \(26873856000000\) | \([2, 2]\) | \(2304\) | \(1.4600\) | |
720.j1 | 720j7 | \([0, 0, 0, -768027, -259067446]\) | \(16778985534208729/81000\) | \(241864704000\) | \([2]\) | \(4608\) | \(1.8066\) | |
720.j2 | 720j8 | \([0, 0, 0, -65307, -874294]\) | \(10316097499609/5859375000\) | \(17496000000000000\) | \([4]\) | \(4608\) | \(1.8066\) |
Rank
sage: E.rank()
The elliptic curves in class 720j have rank \(0\).
Complex multiplication
The elliptic curves in class 720j do not have complex multiplication.Modular form 720.2.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 Cremona numbering.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.