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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 2139.b
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 2139.b1 | 2139c5 | \([1, 0, 0, -144214, -20979607]\) | \(331700532630485560417/2031935135086107\) | \(2031935135086107\) | \([2]\) | \(10368\) | \(1.7759\) | |
| 2139.b2 | 2139c3 | \([1, 0, 0, -14479, 115304]\) | \(335692231577164657/188402060277369\) | \(188402060277369\) | \([2, 2]\) | \(5184\) | \(1.4293\) | |
| 2139.b3 | 2139c2 | \([1, 0, 0, -10834, 432419]\) | \(140634771298875937/270168129729\) | \(270168129729\) | \([2, 4]\) | \(2592\) | \(1.0827\) | |
| 2139.b4 | 2139c1 | \([1, 0, 0, -10829, 432840]\) | \(140440148435570257/519777\) | \(519777\) | \([4]\) | \(1296\) | \(0.73615\) | \(\Gamma_0(N)\)-optimal |
| 2139.b5 | 2139c4 | \([1, 0, 0, -7269, 722610]\) | \(-42476766863084497/201372259510953\) | \(-201372259510953\) | \([4]\) | \(5184\) | \(1.4293\) | |
| 2139.b6 | 2139c6 | \([1, 0, 0, 56936, 929435]\) | \(20411931106401081983/12181842687769803\) | \(-12181842687769803\) | \([2]\) | \(10368\) | \(1.7759\) |
Rank
sage: E.rank()
The elliptic curves in class 2139.b have rank \(0\).
Complex multiplication
The elliptic curves in class 2139.b do not have complex multiplication.Modular form 2139.2.a.b
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
