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SageMath
E = EllipticCurve("dp1")
E.isogeny_class()
Elliptic curves in class 22050.dp
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
22050.dp1 | 22050dd2 | \([1, -1, 1, -2197880, -1253607253]\) | \(68971442301/400\) | \(6809671181250000\) | \([2]\) | \(344064\) | \(2.2283\) | |
22050.dp2 | 22050dd1 | \([1, -1, 1, -139880, -18807253]\) | \(17779581/1280\) | \(21790947780000000\) | \([2]\) | \(172032\) | \(1.8817\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 22050.dp have rank \(1\).
Complex multiplication
The elliptic curves in class 22050.dp do not have complex multiplication.Modular form 22050.2.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 LMFDB 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 LMFDB labels.