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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 4046.f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 4046.f1 | 4046b6 | \([1, 1, 0, -789120, -270141952]\) | \(2251439055699625/25088\) | \(605563331072\) | \([2]\) | \(27648\) | \(1.8297\) | |
| 4046.f2 | 4046b5 | \([1, 1, 0, -49280, -4243456]\) | \(-548347731625/1835008\) | \(-44292632215552\) | \([2]\) | \(13824\) | \(1.4831\) | |
| 4046.f3 | 4046b4 | \([1, 1, 0, -10265, -332419]\) | \(4956477625/941192\) | \(22718086842248\) | \([2]\) | \(9216\) | \(1.2804\) | |
| 4046.f4 | 4046b2 | \([1, 1, 0, -3040, 63222]\) | \(128787625/98\) | \(2365481762\) | \([2]\) | \(3072\) | \(0.73110\) | |
| 4046.f5 | 4046b1 | \([1, 1, 0, -150, 1376]\) | \(-15625/28\) | \(-675851932\) | \([2]\) | \(1536\) | \(0.38452\) | \(\Gamma_0(N)\)-optimal |
| 4046.f6 | 4046b3 | \([1, 1, 0, 1295, -29547]\) | \(9938375/21952\) | \(-529867914688\) | \([2]\) | \(4608\) | \(0.93383\) |
Rank
sage: E.rank()
The elliptic curves in class 4046.f have rank \(1\).
Complex multiplication
The elliptic curves in class 4046.f do not have complex multiplication.Modular form 4046.2.a.f
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 9 & 18 & 6 \\ 2 & 1 & 6 & 18 & 9 & 3 \\ 3 & 6 & 1 & 3 & 6 & 2 \\ 9 & 18 & 3 & 1 & 2 & 6 \\ 18 & 9 & 6 & 2 & 1 & 3 \\ 6 & 3 & 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
