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SageMath
E = EllipticCurve("be1")
E.isogeny_class()
Elliptic curves in class 422370.be
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
422370.be1 | 422370be4 | \([1, -1, 0, -2835004365, -58099242399899]\) | \(73474353581350183614361/576510977802240\) | \(19772278338663934114037760\) | \([2]\) | \(246343680\) | \(4.0269\) | |
422370.be2 | 422370be3 | \([1, -1, 0, -173423565, -948182513819]\) | \(-16818951115904497561/1592332281446400\) | \(-54611340093506279020953600\) | \([2]\) | \(123171840\) | \(3.6803\) | |
422370.be3 | 422370be2 | \([1, -1, 0, -51992190, 5445040756]\) | \(453198971846635561/261896250564000\) | \(8982110942179112498436000\) | \([2]\) | \(82114560\) | \(3.4776\) | \(\Gamma_0(N)\)-optimal* |
422370.be4 | 422370be1 | \([1, -1, 0, 12987810, 675508756]\) | \(7064514799444439/4094064000000\) | \(-140411850010029936000000\) | \([2]\) | \(41057280\) | \(3.1310\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 422370.be have rank \(1\).
Complex multiplication
The elliptic curves in class 422370.be do not have complex multiplication.Modular form 422370.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.