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SageMath
E = EllipticCurve("o1")
E.isogeny_class()
Elliptic curves in class 29400o
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 29400.cc4 | 29400o1 | \([0, -1, 0, -900783, -328762188]\) | \(2748251600896/2205\) | \(64854011250000\) | \([2]\) | \(294912\) | \(1.9557\) | \(\Gamma_0(N)\)-optimal |
| 29400.cc3 | 29400o2 | \([0, -1, 0, -906908, -324058188]\) | \(175293437776/4862025\) | \(2288049516900000000\) | \([2, 2]\) | \(589824\) | \(2.3023\) | |
| 29400.cc5 | 29400o3 | \([0, -1, 0, 195592, -1062733188]\) | \(439608956/259416045\) | \(-488320612451280000000\) | \([2]\) | \(1179648\) | \(2.6488\) | |
| 29400.cc2 | 29400o4 | \([0, -1, 0, -2107408, 715574812]\) | \(549871953124/200930625\) | \(378228593610000000000\) | \([2, 2]\) | \(1179648\) | \(2.6488\) | |
| 29400.cc6 | 29400o5 | \([0, -1, 0, 6467592, 5054524812]\) | \(7947184069438/7533176175\) | \(-28360660602002400000000\) | \([2]\) | \(2359296\) | \(2.9954\) | |
| 29400.cc1 | 29400o6 | \([0, -1, 0, -29890408, 62893928812]\) | \(784478485879202/221484375\) | \(833837287500000000000\) | \([2]\) | \(2359296\) | \(2.9954\) |
Rank
sage: E.rank()
The elliptic curves in class 29400o have rank \(0\).
Complex multiplication
The elliptic curves in class 29400o do not have complex multiplication.Modular form 29400.2.a.o
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
