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SageMath
E = EllipticCurve("dp1")
E.isogeny_class()
Elliptic curves in class 47040dp
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
47040.fs2 | 47040dp1 | \([0, 1, 0, 915, 65043]\) | \(2048/45\) | \(-1859494210560\) | \([2]\) | \(86016\) | \(1.0346\) | \(\Gamma_0(N)\)-optimal |
47040.fs1 | 47040dp2 | \([0, 1, 0, -19665, 999375]\) | \(1272112/75\) | \(49586512281600\) | \([2]\) | \(172032\) | \(1.3811\) |
Rank
sage: E.rank()
The elliptic curves in class 47040dp have rank \(0\).
Complex multiplication
The elliptic curves in class 47040dp do not have complex multiplication.Modular form 47040.2.a.dp
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 Cremona numbering.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.