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SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 479370.z
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
479370.z1 | 479370z4 | \([1, 0, 1, -522279, -145110848]\) | \(26487576322129/44531250\) | \(26488226013281250\) | \([2]\) | \(6193152\) | \(2.0466\) | |
479370.z2 | 479370z2 | \([1, 0, 1, -42909, -724604]\) | \(14688124849/8122500\) | \(4831452424822500\) | \([2, 2]\) | \(3096576\) | \(1.7000\) | |
479370.z3 | 479370z1 | \([1, 0, 1, -26089, 1610012]\) | \(3301293169/22800\) | \(13561971718800\) | \([2]\) | \(1548288\) | \(1.3534\) | \(\Gamma_0(N)\)-optimal* |
479370.z4 | 479370z3 | \([1, 0, 1, 167341, -5686504]\) | \(871257511151/527800050\) | \(-313947778564966050\) | \([2]\) | \(6193152\) | \(2.0466\) |
Rank
sage: E.rank()
The elliptic curves in class 479370.z have rank \(1\).
Complex multiplication
The elliptic curves in class 479370.z do not have complex multiplication.Modular form 479370.2.a.z
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.