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SageMath
E = EllipticCurve("fs1")
E.isogeny_class()
Elliptic curves in class 344850.fs
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
344850.fs1 | 344850fs4 | \([1, 1, 1, -9196063, -10737570469]\) | \(3107086841064961/570\) | \(15777965156250\) | \([2]\) | \(11796480\) | \(2.3682\) | |
344850.fs2 | 344850fs3 | \([1, 1, 1, -665563, -111471469]\) | \(1177918188481/488703750\) | \(13527632875839843750\) | \([2]\) | \(11796480\) | \(2.3682\) | |
344850.fs3 | 344850fs2 | \([1, 1, 1, -574813, -167917969]\) | \(758800078561/324900\) | \(8993440139062500\) | \([2, 2]\) | \(5898240\) | \(2.0216\) | |
344850.fs4 | 344850fs1 | \([1, 1, 1, -30313, -3478969]\) | \(-111284641/123120\) | \(-3408040473750000\) | \([2]\) | \(2949120\) | \(1.6750\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 344850.fs have rank \(0\).
Complex multiplication
The elliptic curves in class 344850.fs do not have complex multiplication.Modular form 344850.2.a.fs
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.