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SageMath
E = EllipticCurve("fn1")
E.isogeny_class()
Elliptic curves in class 22050.fn
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
22050.fn1 | 22050eq4 | \([1, -1, 1, -14817830, 21958278297]\) | \(268498407453697/252\) | \(337704101437500\) | \([2]\) | \(786432\) | \(2.5161\) | |
22050.fn2 | 22050eq6 | \([1, -1, 1, -10077080, -12191879703]\) | \(84448510979617/933897762\) | \(1251512319645643781250\) | \([2]\) | \(1572864\) | \(2.8627\) | |
22050.fn3 | 22050eq3 | \([1, -1, 1, -1146830, 167586297]\) | \(124475734657/63011844\) | \(84441897452142562500\) | \([2, 2]\) | \(786432\) | \(2.5161\) | |
22050.fn4 | 22050eq2 | \([1, -1, 1, -926330, 343104297]\) | \(65597103937/63504\) | \(85101433562250000\) | \([2, 2]\) | \(393216\) | \(2.1695\) | |
22050.fn5 | 22050eq1 | \([1, -1, 1, -44330, 7944297]\) | \(-7189057/16128\) | \(-21613062492000000\) | \([2]\) | \(196608\) | \(1.8229\) | \(\Gamma_0(N)\)-optimal |
22050.fn6 | 22050eq5 | \([1, -1, 1, 4255420, 1291254297]\) | \(6359387729183/4218578658\) | \(-5653298869219462781250\) | \([2]\) | \(1572864\) | \(2.8627\) |
Rank
sage: E.rank()
The elliptic curves in class 22050.fn have rank \(0\).
Complex multiplication
The elliptic curves in class 22050.fn do not have complex multiplication.Modular form 22050.2.a.fn
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.