Show commands:
SageMath
E = EllipticCurve("dz1")
E.isogeny_class()
Elliptic curves in class 25200dz
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 25200.p7 | 25200dz1 | \([0, 0, 0, 755925, 190620250]\) | \(1023887723039/928972800\) | \(-43342154956800000000\) | \([2]\) | \(589824\) | \(2.4553\) | \(\Gamma_0(N)\)-optimal |
| 25200.p6 | 25200dz2 | \([0, 0, 0, -3852075, 1706652250]\) | \(135487869158881/51438240000\) | \(2399902525440000000000\) | \([2, 2]\) | \(1179648\) | \(2.8019\) | |
| 25200.p5 | 25200dz3 | \([0, 0, 0, -27180075, -53324099750]\) | \(47595748626367201/1215506250000\) | \(56710659600000000000000\) | \([2, 2]\) | \(2359296\) | \(3.1485\) | |
| 25200.p4 | 25200dz4 | \([0, 0, 0, -54252075, 153763452250]\) | \(378499465220294881/120530818800\) | \(5623485881932800000000\) | \([2]\) | \(2359296\) | \(3.1485\) | |
| 25200.p8 | 25200dz5 | \([0, 0, 0, 4571925, -170457227750]\) | \(226523624554079/269165039062500\) | \(-12558164062500000000000000\) | \([2]\) | \(4718592\) | \(3.4950\) | |
| 25200.p2 | 25200dz6 | \([0, 0, 0, -432180075, -3458159099750]\) | \(191342053882402567201/129708022500\) | \(6051657497760000000000\) | \([2, 2]\) | \(4718592\) | \(3.4950\) | |
| 25200.p3 | 25200dz7 | \([0, 0, 0, -429480075, -3503500199750]\) | \(-187778242790732059201/4984939585440150\) | \(-232577341298295638400000000\) | \([2]\) | \(9437184\) | \(3.8416\) | |
| 25200.p1 | 25200dz8 | \([0, 0, 0, -6914880075, -221322257999750]\) | \(783736670177727068275201/360150\) | \(16803158400000000\) | \([2]\) | \(9437184\) | \(3.8416\) |
Rank
sage: E.rank()
The elliptic curves in class 25200dz have rank \(1\).
Complex multiplication
The elliptic curves in class 25200dz do not have complex multiplication.Modular form 25200.2.a.dz
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 2 & 2 & 4 & 4 \\ 4 & 2 & 4 & 1 & 8 & 8 & 16 & 16 \\ 8 & 4 & 2 & 8 & 1 & 4 & 8 & 8 \\ 8 & 4 & 2 & 8 & 4 & 1 & 2 & 2 \\ 16 & 8 & 4 & 16 & 8 & 2 & 1 & 4 \\ 16 & 8 & 4 & 16 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
