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SageMath
E = EllipticCurve("ok1")
E.isogeny_class()
Elliptic curves in class 479808.ok
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
479808.ok1 | 479808ok4 | \([0, 0, 0, -319932136524, -69652211962529392]\) | \(322159999717985454060440834/4250799\) | \(47785514287848357888\) | \([2]\) | \(849346560\) | \(4.7598\) | |
479808.ok2 | 479808ok3 | \([0, 0, 0, -20047197324, -1082434979531248]\) | \(79260902459030376659234/842751810121431609\) | \(9473825665167357125685240004608\) | \([2]\) | \(849346560\) | \(4.7598\) | \(\Gamma_0(N)\)-optimal* |
479808.ok3 | 479808ok2 | \([0, 0, 0, -19995759084, -1088315748908080]\) | \(157304700372188331121828/18069292138401\) | \(101563308174635755930976256\) | \([2, 2]\) | \(424673280\) | \(4.4133\) | \(\Gamma_0(N)\)-optimal* |
479808.ok4 | 479808ok1 | \([0, 0, 0, -1246520604, -17096757591760]\) | \(-152435594466395827792/1646846627220711\) | \(-2314135914064337196351995904\) | \([2]\) | \(212336640\) | \(4.0667\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 479808.ok have rank \(1\).
Complex multiplication
The elliptic curves in class 479808.ok do not have complex multiplication.Modular form 479808.2.a.ok
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.