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SageMath
E = EllipticCurve("hw1")
E.isogeny_class()
Elliptic curves in class 331200.hw
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
331200.hw1 | 331200hw3 | \([0, 0, 0, -950700, 351106000]\) | \(63649751618/1164375\) | \(1738402560000000000\) | \([2]\) | \(4718592\) | \(2.2948\) | |
331200.hw2 | 331200hw2 | \([0, 0, 0, -122700, -8246000]\) | \(273671716/119025\) | \(88851686400000000\) | \([2, 2]\) | \(2359296\) | \(1.9483\) | |
331200.hw3 | 331200hw1 | \([0, 0, 0, -104700, -13034000]\) | \(680136784/345\) | \(64385280000000\) | \([2]\) | \(1179648\) | \(1.6017\) | \(\Gamma_0(N)\)-optimal |
331200.hw4 | 331200hw4 | \([0, 0, 0, 417300, -61166000]\) | \(5382838942/4197615\) | \(-6267005614080000000\) | \([2]\) | \(4718592\) | \(2.2948\) |
Rank
sage: E.rank()
The elliptic curves in class 331200.hw have rank \(2\).
Complex multiplication
The elliptic curves in class 331200.hw do not have complex multiplication.Modular form 331200.2.a.hw
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.