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SageMath
E = EllipticCurve("o1")
E.isogeny_class()
Elliptic curves in class 1680.o
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1680.o1 | 1680r3 | \([0, 1, 0, -5976, -179820]\) | \(5763259856089/5670\) | \(23224320\) | \([2]\) | \(1536\) | \(0.70673\) | |
1680.o2 | 1680r2 | \([0, 1, 0, -376, -2860]\) | \(1439069689/44100\) | \(180633600\) | \([2, 2]\) | \(768\) | \(0.36016\) | |
1680.o3 | 1680r1 | \([0, 1, 0, -56, 84]\) | \(4826809/1680\) | \(6881280\) | \([2]\) | \(384\) | \(0.013583\) | \(\Gamma_0(N)\)-optimal |
1680.o4 | 1680r4 | \([0, 1, 0, 104, -9196]\) | \(30080231/9003750\) | \(-36879360000\) | \([2]\) | \(1536\) | \(0.70673\) |
Rank
sage: E.rank()
The elliptic curves in class 1680.o have rank \(0\).
Complex multiplication
The elliptic curves in class 1680.o do not have complex multiplication.Modular form 1680.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.