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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 13357.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
13357.b1 | 13357a3 | \([0, -1, 1, -676273, 214283446]\) | \(727057727488000/37\) | \(1740697597\) | \([]\) | \(42768\) | \(1.6943\) | |
13357.b2 | 13357a2 | \([0, -1, 1, -8423, 290949]\) | \(1404928000/50653\) | \(2383015010293\) | \([]\) | \(14256\) | \(1.1450\) | |
13357.b3 | 13357a1 | \([0, -1, 1, -1203, -15540]\) | \(4096000/37\) | \(1740697597\) | \([]\) | \(4752\) | \(0.59569\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 13357.b have rank \(0\).
Complex multiplication
The elliptic curves in class 13357.b do not have complex multiplication.Modular form 13357.2.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.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.