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SageMath
E = EllipticCurve("u1")
E.isogeny_class()
Elliptic curves in class 27930u
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
27930.o3 | 27930u1 | \([1, 1, 0, -144341432, 667414283376]\) | \(2826887369998878529467769/2262330000\) | \(266160862170000\) | \([2]\) | \(2703360\) | \(2.9715\) | \(\Gamma_0(N)\)-optimal |
27930.o2 | 27930u2 | \([1, 1, 0, -144342412, 667404766204]\) | \(2826944949483509435147449/79970891076562500\) | \(9408495364266501562500\) | \([2, 2]\) | \(5406720\) | \(3.3181\) | |
27930.o4 | 27930u3 | \([1, 1, 0, -138523662, 723684879954]\) | \(-2498661176703400098047449/477389682289643523750\) | \(-56164418731694270925663750\) | \([2]\) | \(10813440\) | \(3.6647\) | |
27930.o1 | 27930u4 | \([1, 1, 0, -150176842, 610515573046]\) | \(3183789741641358436216729/473551070251464843750\) | \(55712809864014587402343750\) | \([2]\) | \(10813440\) | \(3.6647\) |
Rank
sage: E.rank()
The elliptic curves in class 27930u have rank \(0\).
Complex multiplication
The elliptic curves in class 27930u do not have complex multiplication.Modular form 27930.2.a.u
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 Cremona 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 Cremona labels.