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SageMath
E = EllipticCurve("et1")
E.isogeny_class()
Elliptic curves in class 20160et
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
20160.ea4 | 20160et1 | \([0, 0, 0, -14401407, 20777112344]\) | \(7079962908642659949376/100085966990454375\) | \(4669610875906639320000\) | \([2]\) | \(1720320\) | \(2.9627\) | \(\Gamma_0(N)\)-optimal |
20160.ea2 | 20160et2 | \([0, 0, 0, -229635012, 1339384270016]\) | \(448487713888272974160064/91549016015625\) | \(273363897038400000000\) | \([2, 2]\) | \(3440640\) | \(3.3093\) | |
20160.ea1 | 20160et3 | \([0, 0, 0, -3674160012, 85720602100016]\) | \(229625675762164624948320008/9568125\) | \(228562145280000\) | \([4]\) | \(6881280\) | \(3.6559\) | |
20160.ea3 | 20160et4 | \([0, 0, 0, -228847692, 1349024531024]\) | \(-55486311952875723077768/801237030029296875\) | \(-19139847615000000000000000\) | \([2]\) | \(6881280\) | \(3.6559\) |
Rank
sage: E.rank()
The elliptic curves in class 20160et have rank \(0\).
Complex multiplication
The elliptic curves in class 20160et do not have complex multiplication.Modular form 20160.2.a.et
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.