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SageMath
E = EllipticCurve("w1")
E.isogeny_class()
Elliptic curves in class 450840.w
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
450840.w1 | 450840w4 | \([0, -1, 0, -9449240, -11176780980]\) | \(1887517194957938/21849165\) | \(1080085970493204480\) | \([2]\) | \(14155776\) | \(2.6115\) | |
450840.w2 | 450840w2 | \([0, -1, 0, -605840, -164979300]\) | \(994958062276/98903025\) | \(2444573276412134400\) | \([2, 2]\) | \(7077888\) | \(2.2649\) | |
450840.w3 | 450840w1 | \([0, -1, 0, -137660, 16861812]\) | \(46689225424/7249905\) | \(44798741038321920\) | \([2]\) | \(3538944\) | \(1.9183\) | \(\Gamma_0(N)\)-optimal* |
450840.w4 | 450840w3 | \([0, -1, 0, 746680, -798499668]\) | \(931329171502/6107473125\) | \(-301915246531242240000\) | \([2]\) | \(14155776\) | \(2.6115\) |
Rank
sage: E.rank()
The elliptic curves in class 450840.w have rank \(0\).
Complex multiplication
The elliptic curves in class 450840.w do not have complex multiplication.Modular form 450840.2.a.w
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.