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SageMath
E = EllipticCurve("ej1")
E.isogeny_class()
Elliptic curves in class 239400.ej
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
239400.ej1 | 239400ej3 | \([0, 0, 0, -727275, 198103750]\) | \(1823652903746/328593657\) | \(7665432830496000000\) | \([2]\) | \(5242880\) | \(2.3426\) | |
239400.ej2 | 239400ej2 | \([0, 0, 0, -214275, -35311250]\) | \(93280467172/7800849\) | \(90989102736000000\) | \([2, 2]\) | \(2621440\) | \(1.9960\) | |
239400.ej3 | 239400ej1 | \([0, 0, 0, -209775, -36980750]\) | \(350104249168/2793\) | \(8144388000000\) | \([2]\) | \(1310720\) | \(1.6495\) | \(\Gamma_0(N)\)-optimal |
239400.ej4 | 239400ej4 | \([0, 0, 0, 226725, -161878250]\) | \(55251546334/517244049\) | \(-12066269175072000000\) | \([2]\) | \(5242880\) | \(2.3426\) |
Rank
sage: E.rank()
The elliptic curves in class 239400.ej have rank \(0\).
Complex multiplication
The elliptic curves in class 239400.ej do not have complex multiplication.Modular form 239400.2.a.ej
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.