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SageMath
E = EllipticCurve("es1")
E.isogeny_class()
Elliptic curves in class 414960.es
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
414960.es1 | 414960es2 | \([0, 1, 0, -415416, -40690476]\) | \(1935594897227176249/946696265563230\) | \(3877667903746990080\) | \([2]\) | \(8478720\) | \(2.2601\) | |
414960.es2 | 414960es1 | \([0, 1, 0, -221016, 39480084]\) | \(291498868418706649/3685655528100\) | \(15096445043097600\) | \([2]\) | \(4239360\) | \(1.9136\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 414960.es have rank \(2\).
Complex multiplication
The elliptic curves in class 414960.es do not have complex multiplication.Modular form 414960.2.a.es
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.