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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 19110.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
19110.b1 | 19110c4 | \([1, 1, 0, -42756298, 107590373332]\) | \(73474353581350183614361/576510977802240\) | \(67825940027455733760\) | \([2]\) | \(1555200\) | \(2.9783\) | |
19110.b2 | 19110c3 | \([1, 1, 0, -2615498, 1755140052]\) | \(-16818951115904497561/1592332281446400\) | \(-187336300579887513600\) | \([2]\) | \(777600\) | \(2.6318\) | |
19110.b3 | 19110c2 | \([1, 1, 0, -784123, -10387523]\) | \(453198971846635561/261896250564000\) | \(30811831982604036000\) | \([2]\) | \(518400\) | \(2.4290\) | |
19110.b4 | 19110c1 | \([1, 1, 0, 195877, -1175523]\) | \(7064514799444439/4094064000000\) | \(-481662535536000000\) | \([2]\) | \(259200\) | \(2.0825\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 19110.b have rank \(0\).
Complex multiplication
The elliptic curves in class 19110.b do not have complex multiplication.Modular form 19110.2.a.b
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.