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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.
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 147.c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
147.c1 | 147b2 | \([0, 1, 1, -44704, -3655907]\) | \(-1713910976512/1594323\) | \(-9190954824723\) | \([]\) | \(546\) | \(1.4103\) | |
147.c2 | 147b1 | \([0, 1, 1, -114, 473]\) | \(-28672/3\) | \(-17294403\) | \([]\) | \(42\) | \(0.12787\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 147.c have rank \(0\).
Complex multiplication
The elliptic curves in class 147.c do not have complex multiplication.Modular form 147.2.a.c
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.