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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 3234.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
3234.b1 | 3234c2 | \([1, 1, 0, -22159540, -39589128368]\) | \(10228636028672744397625/167006381634183168\) | \(19648133792880015532032\) | \([2]\) | \(399360\) | \(3.0769\) | |
3234.b2 | 3234c1 | \([1, 1, 0, -82100, -1735149744]\) | \(-520203426765625/11054534935707648\) | \(-1300554980651069079552\) | \([2]\) | \(199680\) | \(2.7303\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 3234.b have rank \(0\).
Complex multiplication
The elliptic curves in class 3234.b do not have complex multiplication.Modular form 3234.2.a.b
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.