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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 156.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
156.b1 | 156b4 | \([0, 1, 0, -748, -7564]\) | \(181037698000/14480427\) | \(3706989312\) | \([2]\) | \(72\) | \(0.57946\) | |
156.b2 | 156b3 | \([0, 1, 0, -733, -7888]\) | \(2725888000000/19773\) | \(316368\) | \([2]\) | \(36\) | \(0.23288\) | |
156.b3 | 156b2 | \([0, 1, 0, -148, 644]\) | \(1409938000/4563\) | \(1168128\) | \([6]\) | \(24\) | \(0.030152\) | |
156.b4 | 156b1 | \([0, 1, 0, -13, -4]\) | \(16384000/9477\) | \(151632\) | \([6]\) | \(12\) | \(-0.31642\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 156.b have rank \(0\).
Complex multiplication
The elliptic curves in class 156.b do not have complex multiplication.Modular form 156.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.