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SageMath
E = EllipticCurve("s1")
E.isogeny_class()
Elliptic curves in class 21675.s
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
21675.s1 | 21675c8 | \([1, 1, 0, -15606150, -23736178125]\) | \(1114544804970241/405\) | \(152745553828125\) | \([2]\) | \(491520\) | \(2.5122\) | |
21675.s2 | 21675c6 | \([1, 1, 0, -975525, -371070000]\) | \(272223782641/164025\) | \(61861949300390625\) | \([2, 2]\) | \(245760\) | \(2.1656\) | |
21675.s3 | 21675c7 | \([1, 1, 0, -794900, -512499375]\) | \(-147281603041/215233605\) | \(-81175249871972578125\) | \([2]\) | \(491520\) | \(2.5122\) | |
21675.s4 | 21675c4 | \([1, 1, 0, -578150, 168962625]\) | \(56667352321/15\) | \(5657242734375\) | \([2]\) | \(122880\) | \(1.8190\) | |
21675.s5 | 21675c3 | \([1, 1, 0, -72400, -3498125]\) | \(111284641/50625\) | \(19093194228515625\) | \([2, 2]\) | \(122880\) | \(1.8190\) | |
21675.s6 | 21675c2 | \([1, 1, 0, -36275, 2607000]\) | \(13997521/225\) | \(84858641015625\) | \([2, 2]\) | \(61440\) | \(1.4725\) | |
21675.s7 | 21675c1 | \([1, 1, 0, -150, 114375]\) | \(-1/15\) | \(-5657242734375\) | \([2]\) | \(30720\) | \(1.1259\) | \(\Gamma_0(N)\)-optimal |
21675.s8 | 21675c5 | \([1, 1, 0, 252725, -25931750]\) | \(4733169839/3515625\) | \(-1325916265869140625\) | \([2]\) | \(245760\) | \(2.1656\) |
Rank
sage: E.rank()
The elliptic curves in class 21675.s have rank \(1\).
Complex multiplication
The elliptic curves in class 21675.s do not have complex multiplication.Modular form 21675.2.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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 16 & 4 & 8 & 16 & 8 \\ 2 & 1 & 2 & 8 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 16 & 4 & 8 & 16 & 8 \\ 16 & 8 & 16 & 1 & 4 & 2 & 4 & 8 \\ 4 & 2 & 4 & 4 & 1 & 2 & 4 & 2 \\ 8 & 4 & 8 & 2 & 2 & 1 & 2 & 4 \\ 16 & 8 & 16 & 4 & 4 & 2 & 1 & 8 \\ 8 & 4 & 8 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.