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SageMath
E = EllipticCurve("hl1")
E.isogeny_class()
Elliptic curves in class 187200.hl
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 187200.hl1 | 187200eh8 | \([0, 0, 0, -1872000300, -31175037502000]\) | \(242970740812818720001/24375\) | \(72783360000000000\) | \([2]\) | \(37748736\) | \(3.5832\) | |
| 187200.hl2 | 187200eh6 | \([0, 0, 0, -117000300, -487107502000]\) | \(59319456301170001/594140625\) | \(1774094400000000000000\) | \([2, 2]\) | \(18874368\) | \(3.2366\) | |
| 187200.hl3 | 187200eh7 | \([0, 0, 0, -114192300, -511598878000]\) | \(-55150149867714721/5950927734375\) | \(-17769375000000000000000000\) | \([2]\) | \(37748736\) | \(3.5832\) | |
| 187200.hl4 | 187200eh4 | \([0, 0, 0, -7488300, -7225918000]\) | \(15551989015681/1445900625\) | \(4317436131840000000000\) | \([2, 2]\) | \(9437184\) | \(2.8900\) | |
| 187200.hl5 | 187200eh2 | \([0, 0, 0, -1656300, 693938000]\) | \(168288035761/27720225\) | \(82772148326400000000\) | \([2, 2]\) | \(4718592\) | \(2.5434\) | |
| 187200.hl6 | 187200eh1 | \([0, 0, 0, -1584300, 767522000]\) | \(147281603041/5265\) | \(15721205760000000\) | \([2]\) | \(2359296\) | \(2.1969\) | \(\Gamma_0(N)\)-optimal |
| 187200.hl7 | 187200eh3 | \([0, 0, 0, 3023700, 3904418000]\) | \(1023887723039/2798036865\) | \(-8354893310300160000000\) | \([2]\) | \(9437184\) | \(2.8900\) | |
| 187200.hl8 | 187200eh5 | \([0, 0, 0, 8711700, -34215118000]\) | \(24487529386319/183539412225\) | \(-548045748273254400000000\) | \([2]\) | \(18874368\) | \(3.2366\) |
Rank
sage: E.rank()
The elliptic curves in class 187200.hl have rank \(0\).
Complex multiplication
The elliptic curves in class 187200.hl do not have complex multiplication.Modular form 187200.2.a.hl
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 16 & 16 & 8 \\ 2 & 1 & 2 & 2 & 4 & 8 & 8 & 4 \\ 4 & 2 & 1 & 4 & 8 & 16 & 16 & 8 \\ 4 & 2 & 4 & 1 & 2 & 4 & 4 & 2 \\ 8 & 4 & 8 & 2 & 1 & 2 & 2 & 4 \\ 16 & 8 & 16 & 4 & 2 & 1 & 4 & 8 \\ 16 & 8 & 16 & 4 & 2 & 4 & 1 & 8 \\ 8 & 4 & 8 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
