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SageMath
sage: E = EllipticCurve("h1")
sage: E.isogeny_class()
Elliptic curves in class 6720.h
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 6720.h1 | 6720bh5 | \([0, -1, 0, -97601, 11765985]\) | \(784478485879202/221484375\) | \(29030400000000\) | \([2]\) | \(32768\) | \(1.5643\) | |
| 6720.h2 | 6720bh3 | \([0, -1, 0, -6881, 135681]\) | \(549871953124/200930625\) | \(13168189440000\) | \([2, 2]\) | \(16384\) | \(1.2177\) | |
| 6720.h3 | 6720bh2 | \([0, -1, 0, -2961, -59535]\) | \(175293437776/4862025\) | \(79659417600\) | \([2, 2]\) | \(8192\) | \(0.87118\) | |
| 6720.h4 | 6720bh1 | \([0, -1, 0, -2941, -60419]\) | \(2748251600896/2205\) | \(2257920\) | \([2]\) | \(4096\) | \(0.52460\) | \(\Gamma_0(N)\)-optimal |
| 6720.h5 | 6720bh4 | \([0, -1, 0, 639, -198495]\) | \(439608956/259416045\) | \(-17001089925120\) | \([2]\) | \(16384\) | \(1.2177\) | |
| 6720.h6 | 6720bh6 | \([0, -1, 0, 21119, 936481]\) | \(7947184069438/7533176175\) | \(-987388467609600\) | \([2]\) | \(32768\) | \(1.5643\) |
Rank
sage: E.rank()
The elliptic curves in class 6720.h have rank \(0\).
Complex multiplication
The elliptic curves in class 6720.h do not have complex multiplication.Modular form 6720.2.a.h
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
