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SageMath
E = EllipticCurve("hf1")
E.isogeny_class()
Elliptic curves in class 103488.hf
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 103488.hf1 | 103488dp4 | \([0, 1, 0, -44321153, 113547531999]\) | \(312196988566716625/25367712678\) | \(782365105547908743168\) | \([2]\) | \(5308416\) | \(3.0542\) | |
| 103488.hf2 | 103488dp3 | \([0, 1, 0, -2580993, 2026172511]\) | \(-61653281712625/21875235228\) | \(-674653680261915475968\) | \([2]\) | \(2654208\) | \(2.7077\) | |
| 103488.hf3 | 103488dp2 | \([0, 1, 0, -1138433, -234870945]\) | \(5290763640625/2291573592\) | \(70674374072784125952\) | \([2]\) | \(1769472\) | \(2.5049\) | |
| 103488.hf4 | 103488dp1 | \([0, 1, 0, 241407, -27067041]\) | \(50447927375/39517632\) | \(-1218762476661768192\) | \([2]\) | \(884736\) | \(2.1584\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 103488.hf have rank \(0\).
Complex multiplication
The elliptic curves in class 103488.hf do not have complex multiplication.Modular form 103488.2.a.hf
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
