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SageMath
E = EllipticCurve("ka1")
E.isogeny_class()
Elliptic curves in class 388080ka
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 388080.ka3 | 388080ka1 | \([0, 0, 0, -114607227, 472243642346]\) | \(473897054735271721/779625\) | \(273880727898624000\) | \([2]\) | \(28311552\) | \(3.0377\) | \(\Gamma_0(N)\)-optimal |
| 388080.ka2 | 388080ka2 | \([0, 0, 0, -114642507, 471938350394]\) | \(474334834335054841/607815140625\) | \(213524262487964736000000\) | \([2, 2]\) | \(56623104\) | \(3.3843\) | |
| 388080.ka4 | 388080ka3 | \([0, 0, 0, -83772507, 731808184394]\) | \(-185077034913624841/551466161890875\) | \(-193728977175094780091904000\) | \([2]\) | \(113246208\) | \(3.7309\) | |
| 388080.ka1 | 388080ka4 | \([0, 0, 0, -146076987, 192529831466]\) | \(981281029968144361/522287841796875\) | \(183478690760211000000000000\) | \([2]\) | \(113246208\) | \(3.7309\) |
Rank
sage: E.rank()
The elliptic curves in class 388080ka have rank \(1\).
Complex multiplication
The elliptic curves in class 388080ka do not have complex multiplication.Modular form 388080.2.a.ka
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.
