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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 5880.p
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
5880.p1 | 5880g4 | \([0, -1, 0, -10600, 423340]\) | \(546718898/405\) | \(97582786560\) | \([2]\) | \(9216\) | \(1.0424\) | |
5880.p2 | 5880g3 | \([0, -1, 0, -6680, -205428]\) | \(136835858/1875\) | \(451772160000\) | \([2]\) | \(9216\) | \(1.0424\) | |
5880.p3 | 5880g2 | \([0, -1, 0, -800, 3900]\) | \(470596/225\) | \(27106329600\) | \([2, 2]\) | \(4608\) | \(0.69578\) | |
5880.p4 | 5880g1 | \([0, -1, 0, 180, 372]\) | \(21296/15\) | \(-451772160\) | \([2]\) | \(2304\) | \(0.34921\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 5880.p have rank \(1\).
Complex multiplication
The elliptic curves in class 5880.p do not have complex multiplication.Modular form 5880.2.a.p
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.