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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 327990.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
327990.b1 | 327990b4 | \([1, 1, 0, -54430378, -154569469772]\) | \(29981943972267024529/4007065140000\) | \(2383495794038129940000\) | \([2]\) | \(32256000\) | \(3.1211\) | |
327990.b2 | 327990b3 | \([1, 1, 0, -21866858, 37791101172]\) | \(1943993954077461649/87266819409120\) | \(51908339334040016087520\) | \([2]\) | \(32256000\) | \(3.1211\) | |
327990.b3 | 327990b2 | \([1, 1, 0, -3701258, -1966130988]\) | \(9427227449071249/2652468249600\) | \(1577749973074128921600\) | \([2, 2]\) | \(16128000\) | \(2.7745\) | |
327990.b4 | 327990b1 | \([1, 1, 0, 604662, -201564972]\) | \(41102915774831/53367275520\) | \(-31744100057528401920\) | \([2]\) | \(8064000\) | \(2.4279\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 327990.b have rank \(1\).
Complex multiplication
The elliptic curves in class 327990.b do not have complex multiplication.Modular form 327990.2.a.b
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.