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SageMath
E = EllipticCurve("hk1")
E.isogeny_class()
Elliptic curves in class 270480.hk
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
270480.hk1 | 270480hk3 | \([0, 1, 0, -1445320, 623962100]\) | \(692895692874169/51420783750\) | \(24779177113205760000\) | \([2]\) | \(7077888\) | \(2.4671\) | |
270480.hk2 | 270480hk2 | \([0, 1, 0, -292840, -49547212]\) | \(5763259856089/1143116100\) | \(550856564936294400\) | \([2, 2]\) | \(3538944\) | \(2.1205\) | |
270480.hk3 | 270480hk1 | \([0, 1, 0, -277160, -56251980]\) | \(4886171981209/270480\) | \(130341689425920\) | \([2]\) | \(1769472\) | \(1.7740\) | \(\Gamma_0(N)\)-optimal |
270480.hk4 | 270480hk4 | \([0, 1, 0, 608760, -293700492]\) | \(51774168853511/107398242630\) | \(-51754171790036459520\) | \([2]\) | \(7077888\) | \(2.4671\) |
Rank
sage: E.rank()
The elliptic curves in class 270480.hk have rank \(1\).
Complex multiplication
The elliptic curves in class 270480.hk do not have complex multiplication.Modular form 270480.2.a.hk
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.