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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 6321.f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 6321.f1 | 6321a3 | \([1, 1, 0, -11981, -503586]\) | \(1616855892553/22851963\) | \(2688510594987\) | \([2]\) | \(11520\) | \(1.1898\) | |
| 6321.f2 | 6321a2 | \([1, 1, 0, -1446, 8415]\) | \(2845178713/1347921\) | \(158581557729\) | \([2, 2]\) | \(5760\) | \(0.84321\) | |
| 6321.f3 | 6321a1 | \([1, 1, 0, -1201, 15520]\) | \(1630532233/1161\) | \(136590489\) | \([2]\) | \(2880\) | \(0.49663\) | \(\Gamma_0(N)\)-optimal |
| 6321.f4 | 6321a4 | \([1, 1, 0, 5169, 70596]\) | \(129784785047/92307627\) | \(-10859900008923\) | \([2]\) | \(11520\) | \(1.1898\) |
Rank
sage: E.rank()
The elliptic curves in class 6321.f have rank \(0\).
Complex multiplication
The elliptic curves in class 6321.f do not have complex multiplication.Modular form 6321.2.a.f
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.
