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SageMath
E = EllipticCurve("be1")
E.isogeny_class()
Elliptic curves in class 34914be
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
34914.z4 | 34914be1 | \([1, 0, 0, -87780684, -525091977072]\) | \(-505304979693115442833/512169554353324032\) | \(-75819475297428143182184448\) | \([4]\) | \(12165120\) | \(3.6614\) | \(\Gamma_0(N)\)-optimal |
34914.z3 | 34914be2 | \([1, 0, 0, -1647865164, -25738865325936]\) | \(3342887139776073669969553/1278380674753560576\) | \(189246219667563195783512064\) | \([2, 2]\) | \(24330240\) | \(4.0080\) | |
34914.z2 | 34914be3 | \([1, 0, 0, -1893659724, -17554545543792]\) | \(5072972674420068408718993/2036482219218784389888\) | \(301472455754745632656392690432\) | \([2]\) | \(48660480\) | \(4.3546\) | |
34914.z1 | 34914be4 | \([1, 0, 0, -26363422284, -1647598439097456]\) | \(13688695234222145601259673233/2003024259937536\) | \(296519477008420226229504\) | \([2]\) | \(48660480\) | \(4.3546\) |
Rank
sage: E.rank()
The elliptic curves in class 34914be have rank \(0\).
Complex multiplication
The elliptic curves in class 34914be do not have complex multiplication.Modular form 34914.2.a.be
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.