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SageMath
E = EllipticCurve("fi1")
E.isogeny_class()
Elliptic curves in class 457776fi
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
457776.fi4 | 457776fi1 | \([0, 0, 0, -243329619, -1142502890222]\) | \(22106889268753393/4969545596928\) | \(358176991493068511130943488\) | \([2]\) | \(148635648\) | \(3.8082\) | \(\Gamma_0(N)\)-optimal |
457776.fi2 | 457776fi2 | \([0, 0, 0, -3652512339, -84958623734510]\) | \(74768347616680342513/5615307472896\) | \(404719887909617757969186816\) | \([2, 2]\) | \(297271296\) | \(4.1548\) | |
457776.fi3 | 457776fi3 | \([0, 0, 0, -3412804179, -96591229264622]\) | \(-60992553706117024753/20624795251201152\) | \(-1486519636282710268046322696192\) | \([2]\) | \(594542592\) | \(4.5013\) | |
457776.fi1 | 457776fi4 | \([0, 0, 0, -58439144019, -5437557752238830]\) | \(306234591284035366263793/1727485056\) | \(124507439994071457202176\) | \([2]\) | \(594542592\) | \(4.5013\) |
Rank
sage: E.rank()
The elliptic curves in class 457776fi have rank \(0\).
Complex multiplication
The elliptic curves in class 457776fi do not have complex multiplication.Modular form 457776.2.a.fi
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.