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SageMath
E = EllipticCurve("s1")
E.isogeny_class()
Elliptic curves in class 405042.s
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
405042.s1 | 405042s3 | \([1, 1, 0, -5990639055, 178464471262629]\) | \(505384091400037554067434625/815656731648\) | \(38373289533960741888\) | \([2]\) | \(209018880\) | \(3.9132\) | \(\Gamma_0(N)\)-optimal* |
405042.s2 | 405042s4 | \([1, 1, 0, -5990581295, 178468084809093]\) | \(-505369473241574671219626625/20303219722982711328\) | \(-955182859004297602194439968\) | \([2]\) | \(418037760\) | \(4.2597\) | |
405042.s3 | 405042s1 | \([1, 1, 0, -74166735, 243331899813]\) | \(959024269496848362625/11151660319506432\) | \(524639684343921578606592\) | \([2]\) | \(69672960\) | \(3.3639\) | \(\Gamma_0(N)\)-optimal* |
405042.s4 | 405042s2 | \([1, 1, 0, -15020495, 620815032741]\) | \(-7966267523043306625/3534510366354604032\) | \(-166284154088785105091592192\) | \([2]\) | \(139345920\) | \(3.7104\) |
Rank
sage: E.rank()
The elliptic curves in class 405042.s have rank \(1\).
Complex multiplication
The elliptic curves in class 405042.s do not have complex multiplication.Modular form 405042.2.a.s
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.