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SageMath
E = EllipticCurve("ip1")
E.isogeny_class()
Elliptic curves in class 270480.ip
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 270480.ip1 | 270480ip3 | \([0, 1, 0, -339300320, 2405493002100]\) | \(8964546681033941529169/31696875000\) | \(15274416729600000000\) | \([2]\) | \(42467328\) | \(3.3234\) | |
| 270480.ip2 | 270480ip4 | \([0, 1, 0, -28271840, 10413167988]\) | \(5186062692284555089/2903809817953800\) | \(1399317795931941339955200\) | \([2]\) | \(42467328\) | \(3.3234\) | |
| 270480.ip3 | 270480ip2 | \([0, 1, 0, -21215840, 37544899188]\) | \(2191574502231419089/4115217960000\) | \(1983083633770659840000\) | \([2, 2]\) | \(21233664\) | \(2.9768\) | |
| 270480.ip4 | 270480ip1 | \([0, 1, 0, -894560, 974723700]\) | \(-164287467238609/757170892800\) | \(-364873311711343411200\) | \([2]\) | \(10616832\) | \(2.6302\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 270480.ip have rank \(0\).
Complex multiplication
The elliptic curves in class 270480.ip do not have complex multiplication.Modular form 270480.2.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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
