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SageMath
E = EllipticCurve("ii1")
E.isogeny_class()
Elliptic curves in class 487872ii
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
487872.ii4 | 487872ii1 | \([0, 0, 0, 5365140, 2844315056]\) | \(50447927375/39517632\) | \(-13378717204802281930752\) | \([2]\) | \(17694720\) | \(2.9337\) | \(\Gamma_0(N)\)-optimal* |
487872.ii3 | 487872ii2 | \([0, 0, 0, -25301100, 24568279472]\) | \(5290763640625/2291573592\) | \(775813567001205053325312\) | \([2]\) | \(35389440\) | \(3.2802\) | \(\Gamma_0(N)\)-optimal* |
487872.ii2 | 487872ii3 | \([0, 0, 0, -57361260, -212377490512]\) | \(-61653281712625/21875235228\) | \(-7405873558009259473502208\) | \([2]\) | \(53084160\) | \(3.4830\) | |
487872.ii1 | 487872ii4 | \([0, 0, 0, -985015020, -11898217435984]\) | \(312196988566716625/25367712678\) | \(8588253821778581515665408\) | \([2]\) | \(106168320\) | \(3.8295\) |
Rank
sage: E.rank()
The elliptic curves in class 487872ii have rank \(1\).
Complex multiplication
The elliptic curves in class 487872ii do not have complex multiplication.Modular form 487872.2.a.ii
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.