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SageMath
E = EllipticCurve("qn1")
E.isogeny_class()
Elliptic curves in class 235200qn
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 235200.qn4 | 235200qn1 | \([0, 1, 0, -3603133, -2633700637]\) | \(2748251600896/2205\) | \(4150656720000000\) | \([2]\) | \(4718592\) | \(2.3023\) | \(\Gamma_0(N)\)-optimal |
| 235200.qn3 | 235200qn2 | \([0, 1, 0, -3627633, -2596093137]\) | \(175293437776/4862025\) | \(146435169081600000000\) | \([2, 2]\) | \(9437184\) | \(2.6488\) | |
| 235200.qn2 | 235200qn3 | \([0, 1, 0, -8429633, 5716168863]\) | \(549871953124/200930625\) | \(24206629991040000000000\) | \([2, 2]\) | \(18874368\) | \(2.9954\) | |
| 235200.qn5 | 235200qn4 | \([0, 1, 0, 782367, -8501083137]\) | \(439608956/259416045\) | \(-31252519196881920000000\) | \([2]\) | \(18874368\) | \(2.9954\) | |
| 235200.qn1 | 235200qn5 | \([0, 1, 0, -119561633, 503031868863]\) | \(784478485879202/221484375\) | \(53365586400000000000000\) | \([2]\) | \(37748736\) | \(3.3420\) | |
| 235200.qn6 | 235200qn6 | \([0, 1, 0, 25870367, 40462068863]\) | \(7947184069438/7533176175\) | \(-1815082278528153600000000\) | \([2]\) | \(37748736\) | \(3.3420\) |
Rank
sage: E.rank()
The elliptic curves in class 235200qn have rank \(1\).
Complex multiplication
The elliptic curves in class 235200qn do not have complex multiplication.Modular form 235200.2.a.qn
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
