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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 840i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
840.i3 | 840i1 | \([0, 1, 0, -36, -96]\) | \(20720464/105\) | \(26880\) | \([2]\) | \(64\) | \(-0.30352\) | \(\Gamma_0(N)\)-optimal |
840.i2 | 840i2 | \([0, 1, 0, -56, 0]\) | \(19307236/11025\) | \(11289600\) | \([2, 2]\) | \(128\) | \(0.043052\) | |
840.i1 | 840i3 | \([0, 1, 0, -656, 6240]\) | \(15267472418/36015\) | \(73758720\) | \([2]\) | \(256\) | \(0.38963\) | |
840.i4 | 840i4 | \([0, 1, 0, 224, 224]\) | \(604223422/354375\) | \(-725760000\) | \([2]\) | \(256\) | \(0.38963\) |
Rank
sage: E.rank()
The elliptic curves in class 840i have rank \(0\).
Complex multiplication
The elliptic curves in class 840i do not have complex multiplication.Modular form 840.2.a.i
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.