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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 486720k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
486720.k4 | 486720k1 | \([0, 0, 0, -34983, -19456632]\) | \(-21024576/714025\) | \(-160798144892673600\) | \([2]\) | \(5505024\) | \(1.9819\) | \(\Gamma_0(N)\)-optimal* |
486720.k3 | 486720k2 | \([0, 0, 0, -1320228, -580851648]\) | \(17657244864/105625\) | \(1522349300759040000\) | \([2, 2]\) | \(11010048\) | \(2.3284\) | \(\Gamma_0(N)\)-optimal* |
486720.k2 | 486720k3 | \([0, 0, 0, -2111148, 196464528]\) | \(9024895368/5078125\) | \(585518961830400000000\) | \([2]\) | \(22020096\) | \(2.6750\) | \(\Gamma_0(N)\)-optimal* |
486720.k1 | 486720k4 | \([0, 0, 0, -21093228, -37287448848]\) | \(9001508089608/325\) | \(37473213557145600\) | \([2]\) | \(22020096\) | \(2.6750\) |
Rank
sage: E.rank()
The elliptic curves in class 486720k have rank \(1\).
Complex multiplication
The elliptic curves in class 486720k do not have complex multiplication.Modular form 486720.2.a.k
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.