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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 130b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
130.b4 | 130b1 | \([1, -1, 1, -7, -1]\) | \(33076161/16640\) | \(16640\) | \([4]\) | \(8\) | \(-0.49736\) | \(\Gamma_0(N)\)-optimal |
130.b2 | 130b2 | \([1, -1, 1, -87, -289]\) | \(72043225281/67600\) | \(67600\) | \([2, 2]\) | \(16\) | \(-0.15079\) | |
130.b1 | 130b3 | \([1, -1, 1, -1387, -19529]\) | \(294889639316481/260\) | \(260\) | \([2]\) | \(32\) | \(0.19579\) | |
130.b3 | 130b4 | \([1, -1, 1, -67, -441]\) | \(-32798729601/71402500\) | \(-71402500\) | \([4]\) | \(32\) | \(0.19579\) |
Rank
sage: E.rank()
The elliptic curves in class 130b have rank \(0\).
Complex multiplication
The elliptic curves in class 130b do not have complex multiplication.Modular form 130.2.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 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.