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SageMath
E = EllipticCurve("kr1")
E.isogeny_class()
Elliptic curves in class 433200.kr
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
433200.kr1 | 433200kr3 | \([0, 1, 0, -71114115008, 7299286627943988]\) | \(13209596798923694545921/92340\) | \(278029865698560000000\) | \([2]\) | \(796262400\) | \(4.4576\) | \(\Gamma_0(N)\)-optimal* |
433200.kr2 | 433200kr4 | \([0, 1, 0, -4499507008, 111089526535988]\) | \(3345930611358906241/165622259047500\) | \(498678085766390934240000000000\) | \([2]\) | \(796262400\) | \(4.4576\) | |
433200.kr3 | 433200kr2 | \([0, 1, 0, -4444635008, 114050090423988]\) | \(3225005357698077121/8526675600\) | \(25673277798605030400000000\) | \([2, 2]\) | \(398131200\) | \(4.1110\) | \(\Gamma_0(N)\)-optimal* |
433200.kr4 | 433200kr1 | \([0, 1, 0, -274363008, 1828070903988]\) | \(-758575480593601/40535043840\) | \(-122048438324891074560000000\) | \([2]\) | \(199065600\) | \(3.7644\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 433200.kr have rank \(0\).
Complex multiplication
The elliptic curves in class 433200.kr do not have complex multiplication.Modular form 433200.2.a.kr
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.