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SageMath
E = EllipticCurve("s1")
E.isogeny_class()
Elliptic curves in class 99099.s
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
99099.s1 | 99099cd4 | \([1, -1, 1, -10798184, -13654893392]\) | \(107818231938348177/4463459\) | \(5764414329444771\) | \([2]\) | \(2334720\) | \(2.5105\) | |
99099.s2 | 99099cd3 | \([1, -1, 1, -1095194, 82959220]\) | \(112489728522417/62811265517\) | \(81118737507559724973\) | \([2]\) | \(2334720\) | \(2.5105\) | |
99099.s3 | 99099cd2 | \([1, -1, 1, -675929, -212538752]\) | \(26444947540257/169338169\) | \(218694821192608761\) | \([2, 2]\) | \(1167360\) | \(2.1639\) | |
99099.s4 | 99099cd1 | \([1, -1, 1, -17084, -7242650]\) | \(-426957777/17320303\) | \(-22368616537874607\) | \([2]\) | \(583680\) | \(1.8173\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 99099.s have rank \(1\).
Complex multiplication
The elliptic curves in class 99099.s do not have complex multiplication.Modular form 99099.2.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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.