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SageMath
E = EllipticCurve("ef1")
E.isogeny_class()
Elliptic curves in class 28560ef
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 28560.ds5 | 28560ef1 | \([0, 1, 0, 48960, -4033740]\) | \(3168685387909439/3563732336640\) | \(-14597047650877440\) | \([2]\) | \(221184\) | \(1.7881\) | \(\Gamma_0(N)\)-optimal |
| 28560.ds4 | 28560ef2 | \([0, 1, 0, -278720, -38243532]\) | \(584614687782041281/184812061593600\) | \(756990204287385600\) | \([2, 2]\) | \(442368\) | \(2.1347\) | |
| 28560.ds3 | 28560ef3 | \([0, 1, 0, -1758400, 867912500]\) | \(146796951366228945601/5397929064360000\) | \(22109917447618560000\) | \([2, 4]\) | \(884736\) | \(2.4812\) | |
| 28560.ds2 | 28560ef4 | \([0, 1, 0, -4041920, -3128583372]\) | \(1782900110862842086081/328139630024640\) | \(1344059924580925440\) | \([2]\) | \(884736\) | \(2.4812\) | |
| 28560.ds6 | 28560ef5 | \([0, 1, 0, 689600, 3097550900]\) | \(8854313460877886399/1016927675429790600\) | \(-4165335758560422297600\) | \([4]\) | \(1769472\) | \(2.8278\) | |
| 28560.ds1 | 28560ef6 | \([0, 1, 0, -27881280, 56655935028]\) | \(585196747116290735872321/836876053125000\) | \(3427844313600000000\) | \([8]\) | \(1769472\) | \(2.8278\) |
Rank
sage: E.rank()
The elliptic curves in class 28560ef have rank \(0\).
Complex multiplication
The elliptic curves in class 28560ef do not have complex multiplication.Modular form 28560.2.a.ef
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
