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SageMath
E = EllipticCurve("kh1")
E.isogeny_class()
Elliptic curves in class 388080kh
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
388080.kh5 | 388080kh1 | \([0, 0, 0, -2348589747, -51330783002254]\) | \(-4078208988807294650401/880065599546327040\) | \(-309165312813579591754595696640\) | \([2]\) | \(424673280\) | \(4.3797\) | \(\Gamma_0(N)\)-optimal |
388080.kh4 | 388080kh2 | \([0, 0, 0, -39342351027, -3003484724412046]\) | \(19170300594578891358373921/671785075055001600\) | \(235996774535828885015730585600\) | \([2, 2]\) | \(849346560\) | \(4.7263\) | |
388080.kh3 | 388080kh3 | \([0, 0, 0, -41112560307, -2718404203249294]\) | \(21876183941534093095979041/3572502915711058560000\) | \(1255013242231715633131707432960000\) | \([2, 2]\) | \(1698693120\) | \(5.0728\) | |
388080.kh1 | 388080kh4 | \([0, 0, 0, -629472322227, -192226417495801486]\) | \(78519570041710065450485106721/96428056919040\) | \(33874986588214361472368640\) | \([2]\) | \(1698693120\) | \(5.0728\) | |
388080.kh2 | 388080kh5 | \([0, 0, 0, -185139632307, 28064242646581106]\) | \(1997773216431678333214187041/187585177195046990066400\) | \(65898303508958579714951825139302400\) | \([2]\) | \(3397386240\) | \(5.4194\) | |
388080.kh6 | 388080kh6 | \([0, 0, 0, 74591163213, -15255897698663566]\) | \(130650216943167617311657439/361816948816603087500000\) | \(-127105581924579685218298214400000000\) | \([2]\) | \(3397386240\) | \(5.4194\) |
Rank
sage: E.rank()
The elliptic curves in class 388080kh have rank \(1\).
Complex multiplication
The elliptic curves in class 388080kh do not have complex multiplication.Modular form 388080.2.a.kh
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.