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SageMath
E = EllipticCurve("he1")
E.isogeny_class()
Elliptic curves in class 162240.he
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 162240.he1 | 162240k7 | \([0, 1, 0, -1406080225, 20293326089375]\) | \(242970740812818720001/24375\) | \(30842151075840000\) | \([2]\) | \(33030144\) | \(3.5116\) | |
| 162240.he2 | 162240k5 | \([0, 1, 0, -87880225, 317059649375]\) | \(59319456301170001/594140625\) | \(751777432473600000000\) | \([2, 2]\) | \(16515072\) | \(3.1650\) | |
| 162240.he3 | 162240k8 | \([0, 1, 0, -85771105, 333003331103]\) | \(-55150149867714721/5950927734375\) | \(-7529822040000000000000000\) | \([2]\) | \(33030144\) | \(3.5116\) | |
| 162240.he4 | 162240k3 | \([0, 1, 0, -5624545, 4701930143]\) | \(15551989015681/1445900625\) | \(1829525559667752960000\) | \([2, 2]\) | \(8257536\) | \(2.8185\) | |
| 162240.he5 | 162240k2 | \([0, 1, 0, -1244065, -452142625]\) | \(168288035761/27720225\) | \(35074927889488281600\) | \([2, 2]\) | \(4128768\) | \(2.4719\) | |
| 162240.he6 | 162240k1 | \([0, 1, 0, -1189985, -500025057]\) | \(147281603041/5265\) | \(6661904632381440\) | \([2]\) | \(2064384\) | \(2.1253\) | \(\Gamma_0(N)\)-optimal |
| 162240.he7 | 162240k4 | \([0, 1, 0, 2271135, -2540874465]\) | \(1023887723039/2798036865\) | \(-3540409259737424855040\) | \([2]\) | \(8257536\) | \(2.8185\) | |
| 162240.he8 | 162240k6 | \([0, 1, 0, 6543455, 22274955743]\) | \(24487529386319/183539412225\) | \(-232235908931869743513600\) | \([2]\) | \(16515072\) | \(3.1650\) |
Rank
sage: E.rank()
The elliptic curves in class 162240.he have rank \(0\).
Complex multiplication
The elliptic curves in class 162240.he do not have complex multiplication.Modular form 162240.2.a.he
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 16 & 16 & 8 \\ 2 & 1 & 2 & 2 & 4 & 8 & 8 & 4 \\ 4 & 2 & 1 & 4 & 8 & 16 & 16 & 8 \\ 4 & 2 & 4 & 1 & 2 & 4 & 4 & 2 \\ 8 & 4 & 8 & 2 & 1 & 2 & 2 & 4 \\ 16 & 8 & 16 & 4 & 2 & 1 & 4 & 8 \\ 16 & 8 & 16 & 4 & 2 & 4 & 1 & 8 \\ 8 & 4 & 8 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
