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SageMath
E = EllipticCurve("er1")
E.isogeny_class()
Elliptic curves in class 84150.er
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
84150.er1 | 84150eu3 | \([1, -1, 1, -3733777805, 87816360598197]\) | \(505384091400037554067434625/815656731648\) | \(9290839958928000000\) | \([2]\) | \(39813120\) | \(3.7950\) | |
84150.er2 | 84150eu4 | \([1, -1, 1, -3733741805, 87818138638197]\) | \(-505369473241574671219626625/20303219722982711328\) | \(-231266362157099946220500000\) | \([2]\) | \(79626240\) | \(4.1415\) | |
84150.er3 | 84150eu1 | \([1, -1, 1, -46225805, 119759254197]\) | \(959024269496848362625/11151660319506432\) | \(127024380826877952000000\) | \([2]\) | \(13271040\) | \(3.2457\) | \(\Gamma_0(N)\)-optimal |
84150.er4 | 84150eu2 | \([1, -1, 1, -9361805, 305480086197]\) | \(-7966267523043306625/3534510366354604032\) | \(-40260282141757911552000000\) | \([2]\) | \(26542080\) | \(3.5922\) |
Rank
sage: E.rank()
The elliptic curves in class 84150.er have rank \(1\).
Complex multiplication
The elliptic curves in class 84150.er do not have complex multiplication.Modular form 84150.2.a.er
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.