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SageMath
E = EllipticCurve("s1")
E.isogeny_class()
Elliptic curves in class 88200.s
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 88200.s1 | 88200hd6 | \([0, 0, 0, -269013675, -1697867064250]\) | \(784478485879202/221484375\) | \(607867382587500000000000\) | \([2]\) | \(18874368\) | \(3.5447\) | |
| 88200.s2 | 88200hd4 | \([0, 0, 0, -18966675, -19301553250]\) | \(549871953124/200930625\) | \(275728644741690000000000\) | \([2, 2]\) | \(9437184\) | \(3.1982\) | |
| 88200.s3 | 88200hd2 | \([0, 0, 0, -8162175, 8757733250]\) | \(175293437776/4862025\) | \(1667988097820100000000\) | \([2, 2]\) | \(4718592\) | \(2.8516\) | |
| 88200.s4 | 88200hd1 | \([0, 0, 0, -8107050, 8884686125]\) | \(2748251600896/2205\) | \(47278574201250000\) | \([2]\) | \(2359296\) | \(2.5050\) | \(\Gamma_0(N)\)-optimal |
| 88200.s5 | 88200hd3 | \([0, 0, 0, 1760325, 28692035750]\) | \(439608956/259416045\) | \(-355985726476983120000000\) | \([2]\) | \(9437184\) | \(3.1982\) | |
| 88200.s6 | 88200hd5 | \([0, 0, 0, 58208325, -136530378250]\) | \(7947184069438/7533176175\) | \(-20674921578859749600000000\) | \([2]\) | \(18874368\) | \(3.5447\) |
Rank
sage: E.rank()
The elliptic curves in class 88200.s have rank \(0\).
Complex multiplication
The elliptic curves in class 88200.s do not have complex multiplication.Modular form 88200.2.a.s
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
