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SageMath
E = EllipticCurve("fn1")
E.isogeny_class()
Elliptic curves in class 69696fn
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
69696.ea3 | 69696fn1 | \([0, 0, 0, -384780, -86653424]\) | \(18609625/1188\) | \(402198088167456768\) | \([2]\) | \(737280\) | \(2.1286\) | \(\Gamma_0(N)\)-optimal |
69696.ea4 | 69696fn2 | \([0, 0, 0, 312180, -365716208]\) | \(9938375/176418\) | \(-59726416092867330048\) | \([2]\) | \(1474560\) | \(2.4752\) | |
69696.ea1 | 69696fn3 | \([0, 0, 0, -5611980, 5097474448]\) | \(57736239625/255552\) | \(86517277632466255872\) | \([2]\) | \(2211840\) | \(2.6779\) | |
69696.ea2 | 69696fn4 | \([0, 0, 0, -2824140, 10163537296]\) | \(-7357983625/127552392\) | \(-43182936198304719962112\) | \([2]\) | \(4423680\) | \(3.0245\) |
Rank
sage: E.rank()
The elliptic curves in class 69696fn have rank \(0\).
Complex multiplication
The elliptic curves in class 69696fn do not have complex multiplication.Modular form 69696.2.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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.