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SageMath
E = EllipticCurve("cr1")
E.isogeny_class()
Elliptic curves in class 139650.cr
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
139650.cr1 | 139650gd4 | \([1, 0, 1, -3724026, -2766405302]\) | \(3107086841064961/570\) | \(1047811406250\) | \([2]\) | \(2654208\) | \(2.1422\) | |
139650.cr2 | 139650gd3 | \([1, 0, 1, -269526, -28677302]\) | \(1177918188481/488703750\) | \(898367304433593750\) | \([2]\) | \(2654208\) | \(2.1422\) | |
139650.cr3 | 139650gd2 | \([1, 0, 1, -232776, -43230302]\) | \(758800078561/324900\) | \(597252501562500\) | \([2, 2]\) | \(1327104\) | \(1.7956\) | |
139650.cr4 | 139650gd1 | \([1, 0, 1, -12276, -894302]\) | \(-111284641/123120\) | \(-226327263750000\) | \([2]\) | \(663552\) | \(1.4490\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 139650.cr have rank \(0\).
Complex multiplication
The elliptic curves in class 139650.cr do not have complex multiplication.Modular form 139650.2.a.cr
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.