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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 411840.i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
411840.i1 | 411840i4 | \([0, 0, 0, -923790828, -10807085379152]\) | \(912446049969377120252018/17177299425\) | \(1641316519880294400\) | \([2]\) | \(88080384\) | \(3.4832\) | |
411840.i2 | 411840i3 | \([0, 0, 0, -62886828, -136949964752]\) | \(287849398425814280018/81784533026485575\) | \(7814633826065840101785600\) | \([2]\) | \(88080384\) | \(3.4832\) | \(\Gamma_0(N)\)-optimal* |
411840.i3 | 411840i2 | \([0, 0, 0, -57738828, -168849031952]\) | \(445574312599094932036/61129333175625\) | \(2920499372689367040000\) | \([2, 2]\) | \(44040192\) | \(3.1367\) | \(\Gamma_0(N)\)-optimal* |
411840.i4 | 411840i1 | \([0, 0, 0, -3288828, -3125011952]\) | \(-329381898333928144/162600887109375\) | \(-1942094589177600000000\) | \([2]\) | \(22020096\) | \(2.7901\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 411840.i have rank \(0\).
Complex multiplication
The elliptic curves in class 411840.i do not have complex multiplication.Modular form 411840.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.