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SageMath
E = EllipticCurve("s1")
E.isogeny_class()
Elliptic curves in class 275275.s
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
275275.s1 | 275275s3 | \([1, -1, 1, -29994955, 63237095672]\) | \(107818231938348177/4463459\) | \(123551404523421875\) | \([2]\) | \(9338880\) | \(2.7659\) | |
275275.s2 | 275275s4 | \([1, -1, 1, -3042205, -382042328]\) | \(112489728522417/62811265517\) | \(1738656067977531828125\) | \([2]\) | \(9338880\) | \(2.7659\) | |
275275.s3 | 275275s2 | \([1, -1, 1, -1877580, 985227422]\) | \(26444947540257/169338169\) | \(4687389000184515625\) | \([2, 2]\) | \(4669440\) | \(2.4193\) | |
275275.s4 | 275275s1 | \([1, -1, 1, -47455, 33562422]\) | \(-426957777/17320303\) | \(-479437082859109375\) | \([2]\) | \(2334720\) | \(2.0727\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 275275.s have rank \(2\).
Complex multiplication
The elliptic curves in class 275275.s do not have complex multiplication.Modular form 275275.2.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.