Show commands:
SageMath
E = EllipticCurve("eu1")
E.isogeny_class()
Elliptic curves in class 25200.eu
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 25200.eu1 | 25200eh6 | \([0, 0, 0, -9829875, -11862332750]\) | \(2251439055699625/25088\) | \(1170505728000000\) | \([2]\) | \(497664\) | \(2.4603\) | |
| 25200.eu2 | 25200eh5 | \([0, 0, 0, -613875, -185660750]\) | \(-548347731625/1835008\) | \(-85614133248000000\) | \([2]\) | \(248832\) | \(2.1137\) | |
| 25200.eu3 | 25200eh4 | \([0, 0, 0, -127875, -14426750]\) | \(4956477625/941192\) | \(43912253952000000\) | \([2]\) | \(165888\) | \(1.9110\) | |
| 25200.eu4 | 25200eh2 | \([0, 0, 0, -37875, 2835250]\) | \(128787625/98\) | \(4572288000000\) | \([2]\) | \(55296\) | \(1.3617\) | |
| 25200.eu5 | 25200eh1 | \([0, 0, 0, -1875, 63250]\) | \(-15625/28\) | \(-1306368000000\) | \([2]\) | \(27648\) | \(1.0151\) | \(\Gamma_0(N)\)-optimal |
| 25200.eu6 | 25200eh3 | \([0, 0, 0, 16125, -1322750]\) | \(9938375/21952\) | \(-1024192512000000\) | \([2]\) | \(82944\) | \(1.5644\) |
Rank
sage: E.rank()
The elliptic curves in class 25200.eu have rank \(0\).
Complex multiplication
The elliptic curves in class 25200.eu do not have complex multiplication.Modular form 25200.2.a.eu
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 9 & 18 & 6 \\ 2 & 1 & 6 & 18 & 9 & 3 \\ 3 & 6 & 1 & 3 & 6 & 2 \\ 9 & 18 & 3 & 1 & 2 & 6 \\ 18 & 9 & 6 & 2 & 1 & 3 \\ 6 & 3 & 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
