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SageMath
E = EllipticCurve("w1")
E.isogeny_class()
Elliptic curves in class 121680.w
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
121680.w1 | 121680dg2 | \([0, 0, 0, -2613, -50713]\) | \(1000939264/15625\) | \(30800250000\) | \([]\) | \(103680\) | \(0.81350\) | |
121680.w2 | 121680dg1 | \([0, 0, 0, -273, 1703]\) | \(1141504/25\) | \(49280400\) | \([]\) | \(34560\) | \(0.26419\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 121680.w have rank \(1\).
Complex multiplication
The elliptic curves in class 121680.w do not have complex multiplication.Modular form 121680.2.a.w
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.