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Properties

Label 21675.s
Number of curves $8$
Conductor $21675$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("s1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 21675.s have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(3\)\(1 + T\)
\(5\)\(1\)
\(17\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(2\) \( 1 - T + 2 T^{2}\) 1.2.ab
\(7\) \( 1 + 7 T^{2}\) 1.7.a
\(11\) \( 1 - 4 T + 11 T^{2}\) 1.11.ae
\(13\) \( 1 - 2 T + 13 T^{2}\) 1.13.ac
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 - 2 T + 29 T^{2}\) 1.29.ac
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 21675.s do not have complex multiplication.

Modular form 21675.2.a.s

Copy content sage:E.q_eigenform(10)
 
\(q + q^{2} - q^{3} - q^{4} - q^{6} - 3 q^{8} + q^{9} + 4 q^{11} + q^{12} + 2 q^{13} - q^{16} + q^{18} + 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content 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

Copy content sage:E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with LMFDB labels.

Elliptic curves in class 21675.s

Copy content 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 \([1, 1, 0, 252725, -25931750]\) \(4733169839/3515625\) \(-1325916265869140625\) \([2]\) \(245760\) \(2.1656\)