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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 704.c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
704.c1 | 704k3 | \([0, -1, 0, -31281, 2139919]\) | \(-52893159101157376/11\) | \(-704\) | \([]\) | \(400\) | \(0.84328\) | |
704.c2 | 704k2 | \([0, -1, 0, -41, 199]\) | \(-122023936/161051\) | \(-10307264\) | \([]\) | \(80\) | \(0.038564\) | |
704.c3 | 704k1 | \([0, -1, 0, -1, -1]\) | \(-4096/11\) | \(-704\) | \([]\) | \(16\) | \(-0.76616\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 704.c have rank \(1\).
Complex multiplication
The elliptic curves in class 704.c do not have complex multiplication.Modular form 704.2.a.c
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.
\(\left(\begin{array}{rrr} 1 & 5 & 25 \\ 5 & 1 & 5 \\ 25 & 5 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.