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SageMath
E = EllipticCurve("hb1")
E.isogeny_class()
Elliptic curves in class 87360.hb
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 87360.hb1 | 87360dp4 | \([0, 1, 0, -6812065, 5587379135]\) | \(266716694084614489298/51372277695070605\) | \(6733467182048294338560\) | \([2]\) | \(5505024\) | \(2.9056\) | |
| 87360.hb2 | 87360dp2 | \([0, 1, 0, -6459265, 6316193375]\) | \(454771411897393003396/23468066028225\) | \(1538003175225753600\) | \([2, 2]\) | \(2752512\) | \(2.5590\) | |
| 87360.hb3 | 87360dp1 | \([0, 1, 0, -6459185, 6316357743]\) | \(1819018058610682173904/4844385\) | \(79370403840\) | \([2]\) | \(1376256\) | \(2.2125\) | \(\Gamma_0(N)\)-optimal |
| 87360.hb4 | 87360dp3 | \([0, 1, 0, -6107745, 7034489343]\) | \(-192245661431796830258/51935513760073125\) | \(-6807291659560304640000\) | \([4]\) | \(5505024\) | \(2.9056\) |
Rank
sage: E.rank()
The elliptic curves in class 87360.hb have rank \(1\).
Complex multiplication
The elliptic curves in class 87360.hb do not have complex multiplication.Modular form 87360.2.a.hb
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
