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SageMath
E = EllipticCurve("cf1")
E.isogeny_class()
Elliptic curves in class 138600.cf
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 138600.cf1 | 138600bs5 | \([0, 0, 0, -379098075, 2840947577750]\) | \(258286045443018193442/8440380939375\) | \(196897206553740000000000\) | \([2]\) | \(25165824\) | \(3.5637\) | |
| 138600.cf2 | 138600bs4 | \([0, 0, 0, -107100075, -426607650250]\) | \(11647843478225136004/128410942275\) | \(1497785230695600000000\) | \([2]\) | \(12582912\) | \(3.2171\) | |
| 138600.cf3 | 138600bs3 | \([0, 0, 0, -24723075, 40321952750]\) | \(143279368983686884/22699269140625\) | \(264764275256250000000000\) | \([2, 2]\) | \(12582912\) | \(3.2171\) | |
| 138600.cf4 | 138600bs2 | \([0, 0, 0, -6862575, -6311812750]\) | \(12257375872392016/1191317675625\) | \(3473882342122500000000\) | \([2, 2]\) | \(6291456\) | \(2.8705\) | |
| 138600.cf5 | 138600bs1 | \([0, 0, 0, 518550, -473342875]\) | \(84611246065664/580054565475\) | \(-105714944557818750000\) | \([2]\) | \(3145728\) | \(2.5239\) | \(\Gamma_0(N)\)-optimal |
| 138600.cf6 | 138600bs6 | \([0, 0, 0, 43883925, 224257319750]\) | \(400647648358480318/1163177490234375\) | \(-27134604492187500000000000\) | \([2]\) | \(25165824\) | \(3.5637\) |
Rank
sage: E.rank()
The elliptic curves in class 138600.cf have rank \(0\).
Complex multiplication
The elliptic curves in class 138600.cf do not have complex multiplication.Modular form 138600.2.a.cf
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 8 & 4 \\ 8 & 1 & 4 & 2 & 4 & 8 \\ 2 & 4 & 1 & 2 & 4 & 2 \\ 4 & 2 & 2 & 1 & 2 & 4 \\ 8 & 4 & 4 & 2 & 1 & 8 \\ 4 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
