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SageMath
This isogeny class and its quadratic twist by $\Q(\sqrt{-3})$ are the ones of minimal conductor with a $13$-isogeny.
sage: E = EllipticCurve("b1")
sage: E.isogeny_class()
Elliptic curves in class 147.b
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 147.b1 | 147c2 | \([0, -1, 1, -912, 10919]\) | \(-1713910976512/1594323\) | \(-78121827\) | \([]\) | \(78\) | \(0.43739\) | |
| 147.b2 | 147c1 | \([0, -1, 1, -2, -1]\) | \(-28672/3\) | \(-147\) | \([]\) | \(6\) | \(-0.84509\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 147.b have rank \(0\).
Complex multiplication
The elliptic curves in class 147.b do not have complex multiplication.Modular form 147.2.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.
\(\left(\begin{array}{rr} 1 & 13 \\ 13 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
