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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 214774a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
214774.k2 | 214774a1 | \([1, 0, 0, 48657, 97321]\) | \(86058173375/49827568\) | \(-7376268325587952\) | \([2]\) | \(1419264\) | \(1.7342\) | \(\Gamma_0(N)\)-optimal |
214774.k1 | 214774a2 | \([1, 0, 0, -194683, 730005]\) | \(5512402554625/3188422748\) | \(472000996008002972\) | \([2]\) | \(2838528\) | \(2.0808\) |
Rank
sage: E.rank()
The elliptic curves in class 214774a have rank \(2\).
Complex multiplication
The elliptic curves in class 214774a do not have complex multiplication.Modular form 214774.2.a.a
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 Cremona numbering.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.