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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 111090.bc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
111090.bc1 | 111090bd4 | \([1, 0, 1, -1594754358, -67378296632]\) | \(3029968325354577848895529/1753440696000000000000\) | \(259572152241138744000000000000\) | \([2]\) | \(145981440\) | \(4.3335\) | |
111090.bc2 | 111090bd2 | \([1, 0, 1, -1097063223, -13986080583494]\) | \(986396822567235411402169/6336721794060000\) | \(938062244129347019340000\) | \([2]\) | \(48660480\) | \(3.7842\) | |
111090.bc3 | 111090bd1 | \([1, 0, 1, -67248343, -227341860742]\) | \(-227196402372228188089/19338934824115200\) | \(-2862856409000952270412800\) | \([2]\) | \(24330240\) | \(3.4377\) | \(\Gamma_0(N)\)-optimal |
111090.bc4 | 111090bd3 | \([1, 0, 1, 398686922, -8372434744]\) | \(47342661265381757089751/27397579603968000000\) | \(-4055825053121670807552000000\) | \([2]\) | \(72990720\) | \(3.9870\) |
Rank
sage: E.rank()
The elliptic curves in class 111090.bc have rank \(0\).
Complex multiplication
The elliptic curves in class 111090.bc do not have complex multiplication.Modular form 111090.2.a.bc
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.