Show commands:
SageMath
E = EllipticCurve("bi1")
E.isogeny_class()
Elliptic curves in class 28224bi
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 28224.dh5 | 28224bi1 | \([0, 0, 0, -14700, 1388464]\) | \(-15625/28\) | \(-629526072655872\) | \([2]\) | \(73728\) | \(1.5299\) | \(\Gamma_0(N)\)-optimal |
| 28224.dh4 | 28224bi2 | \([0, 0, 0, -296940, 62239408]\) | \(128787625/98\) | \(2203341254295552\) | \([2]\) | \(147456\) | \(1.8765\) | |
| 28224.dh6 | 28224bi3 | \([0, 0, 0, 126420, -29037008]\) | \(9938375/21952\) | \(-493548440962203648\) | \([2]\) | \(221184\) | \(2.0792\) | |
| 28224.dh3 | 28224bi4 | \([0, 0, 0, -1002540, -316696016]\) | \(4956477625/941192\) | \(21160889406254481408\) | \([2]\) | \(442368\) | \(2.4258\) | |
| 28224.dh2 | 28224bi5 | \([0, 0, 0, -4812780, -4075624784]\) | \(-548347731625/1835008\) | \(-41256620697575227392\) | \([2]\) | \(663552\) | \(2.6285\) | |
| 28224.dh1 | 28224bi6 | \([0, 0, 0, -77066220, -260401928528]\) | \(2251439055699625/25088\) | \(564055361099661312\) | \([2]\) | \(1327104\) | \(2.9751\) |
Rank
sage: E.rank()
The elliptic curves in class 28224bi have rank \(1\).
Complex multiplication
The elliptic curves in class 28224bi do not have complex multiplication.Modular form 28224.2.a.bi
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 6 & 9 & 18 \\ 2 & 1 & 6 & 3 & 18 & 9 \\ 3 & 6 & 1 & 2 & 3 & 6 \\ 6 & 3 & 2 & 1 & 6 & 3 \\ 9 & 18 & 3 & 6 & 1 & 2 \\ 18 & 9 & 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
