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SageMath
E = EllipticCurve("hz1")
E.isogeny_class()
Elliptic curves in class 485520.hz
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
485520.hz1 | 485520hz4 | \([0, 1, 0, -7796160, -8380796940]\) | \(530044731605089/26309115\) | \(2601115970115317760\) | \([2]\) | \(18874368\) | \(2.6049\) | |
485520.hz2 | 485520hz3 | \([0, 1, 0, -2478560, 1395356340]\) | \(17032120495489/1339001685\) | \(132383725825244221440\) | \([4]\) | \(18874368\) | \(2.6049\) | \(\Gamma_0(N)\)-optimal* |
485520.hz3 | 485520hz2 | \([0, 1, 0, -513360, -116275500]\) | \(151334226289/28676025\) | \(2835126403412889600\) | \([2, 2]\) | \(9437184\) | \(2.2583\) | \(\Gamma_0(N)\)-optimal* |
485520.hz4 | 485520hz1 | \([0, 1, 0, 64640, -10617100]\) | \(302111711/669375\) | \(-66179421181440000\) | \([2]\) | \(4718592\) | \(1.9118\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 485520.hz have rank \(1\).
Complex multiplication
The elliptic curves in class 485520.hz do not have complex multiplication.Modular form 485520.2.a.hz
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.