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SageMath
E = EllipticCurve("ip1")
E.isogeny_class()
Elliptic curves in class 388080.ip
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 388080.ip1 | 388080ip3 | \([0, 0, 0, -347719827, 2495699522386]\) | \(13235378341603461121/9240\) | \(3245993812131840\) | \([2]\) | \(28311552\) | \(3.1894\) | |
| 388080.ip2 | 388080ip2 | \([0, 0, 0, -21732627, 38994785746]\) | \(3231355012744321/85377600\) | \(29992982824098201600\) | \([2, 2]\) | \(14155776\) | \(2.8428\) | |
| 388080.ip3 | 388080ip4 | \([0, 0, 0, -20885907, 42172864594]\) | \(-2868190647517441/527295615000\) | \(-185237911629248163840000\) | \([2]\) | \(28311552\) | \(3.1894\) | |
| 388080.ip4 | 388080ip1 | \([0, 0, 0, -1411347, 559116754]\) | \(885012508801/127733760\) | \(44872618458910556160\) | \([2]\) | \(7077888\) | \(2.4962\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 388080.ip have rank \(1\).
Complex multiplication
The elliptic curves in class 388080.ip do not have complex multiplication.Modular form 388080.2.a.ip
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
