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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 210.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
210.a1 | 210d3 | \([1, 1, 0, -373, 2623]\) | \(5763259856089/5670\) | \(5670\) | \([2]\) | \(64\) | \(0.013583\) | |
210.a2 | 210d2 | \([1, 1, 0, -23, 33]\) | \(1439069689/44100\) | \(44100\) | \([2, 2]\) | \(32\) | \(-0.33299\) | |
210.a3 | 210d1 | \([1, 1, 0, -3, -3]\) | \(4826809/1680\) | \(1680\) | \([2]\) | \(16\) | \(-0.67956\) | \(\Gamma_0(N)\)-optimal |
210.a4 | 210d4 | \([1, 1, 0, 7, 147]\) | \(30080231/9003750\) | \(-9003750\) | \([2]\) | \(64\) | \(0.013583\) |
Rank
sage: E.rank()
The elliptic curves in class 210.a have rank \(1\).
Complex multiplication
The elliptic curves in class 210.a do not have complex multiplication.Modular form 210.2.a.a
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.