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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 28050.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
28050.e1 | 28050e4 | \([1, 1, 0, -35106175, -80076066875]\) | \(306234591284035366263793/1727485056\) | \(26991954000000\) | \([2]\) | \(1376256\) | \(2.6470\) | |
28050.e2 | 28050e2 | \([1, 1, 0, -2194175, -1251826875]\) | \(74768347616680342513/5615307472896\) | \(87739179264000000\) | \([2, 2]\) | \(688128\) | \(2.3004\) | |
28050.e3 | 28050e3 | \([1, 1, 0, -2050175, -1423042875]\) | \(-60992553706117024753/20624795251201152\) | \(-322262425800018000000\) | \([2]\) | \(1376256\) | \(2.6470\) | |
28050.e4 | 28050e1 | \([1, 1, 0, -146175, -16882875]\) | \(22106889268753393/4969545596928\) | \(77649149952000000\) | \([2]\) | \(344064\) | \(1.9538\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 28050.e have rank \(1\).
Complex multiplication
The elliptic curves in class 28050.e do not have complex multiplication.Modular form 28050.2.a.e
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.