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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 194145.d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
194145.d1 | 194145j3 | \([1, 1, 1, -208051, 36436748]\) | \(157551496201/13125\) | \(82967890018125\) | \([2]\) | \(1290240\) | \(1.7146\) | |
194145.d2 | 194145j2 | \([1, 1, 1, -13906, 481094]\) | \(47045881/11025\) | \(69693027615225\) | \([2, 2]\) | \(645120\) | \(1.3681\) | |
194145.d3 | 194145j1 | \([1, 1, 1, -4661, -117982]\) | \(1771561/105\) | \(663743120145\) | \([2]\) | \(322560\) | \(1.0215\) | \(\Gamma_0(N)\)-optimal |
194145.d4 | 194145j4 | \([1, 1, 1, 32319, 3051204]\) | \(590589719/972405\) | \(-6146925035662845\) | \([2]\) | \(1290240\) | \(1.7146\) |
Rank
sage: E.rank()
The elliptic curves in class 194145.d have rank \(0\).
Complex multiplication
The elliptic curves in class 194145.d do not have complex multiplication.Modular form 194145.2.a.d
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.