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SageMath
E = EllipticCurve("em1")
E.isogeny_class()
Elliptic curves in class 234234.em
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
234234.em1 | 234234em4 | \([1, -1, 1, -21496325, 38364079511]\) | \(312196988566716625/25367712678\) | \(89262480716553101958\) | \([2]\) | \(10616832\) | \(2.8733\) | |
234234.em2 | 234234em3 | \([1, -1, 1, -1251815, 684997499]\) | \(-61653281712625/21875235228\) | \(-76973347478932412508\) | \([2]\) | \(5308416\) | \(2.5268\) | |
234234.em3 | 234234em2 | \([1, -1, 1, -552155, -79067725]\) | \(5290763640625/2291573592\) | \(8063460279722359512\) | \([2]\) | \(3538944\) | \(2.3240\) | |
234234.em4 | 234234em1 | \([1, -1, 1, 117085, -9199069]\) | \(50447927375/39517632\) | \(-139052421049493952\) | \([2]\) | \(1769472\) | \(1.9775\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 234234.em have rank \(1\).
Complex multiplication
The elliptic curves in class 234234.em do not have complex multiplication.Modular form 234234.2.a.em
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.