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Properties

Label 25857.f
Number of curves $6$
Conductor $25857$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("f1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 25857.f have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(3\)\(1\)
\(13\)\(1\)
\(17\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(2\) \( 1 + T + 2 T^{2}\) 1.2.b
\(5\) \( 1 + 2 T + 5 T^{2}\) 1.5.c
\(7\) \( 1 + 7 T^{2}\) 1.7.a
\(11\) \( 1 - 4 T + 11 T^{2}\) 1.11.ae
\(19\) \( 1 - 4 T + 19 T^{2}\) 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 25857.f do not have complex multiplication.

Modular form 25857.2.a.f

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} - q^{4} - 2 q^{5} + 3 q^{8} + 2 q^{10} + 4 q^{11} - q^{16} - q^{17} + 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 25857.f

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
25857.f1 25857h6 \([1, -1, 1, -30684686, 65420629512]\) \(908031902324522977/161726530797\) \(569074221230118107517\) \([2]\) \(1376256\) \(2.9869\)  
25857.f2 25857h4 \([1, -1, 1, -2112701, 802228236]\) \(296380748763217/92608836489\) \(325866765609137895129\) \([2, 2]\) \(688128\) \(2.6404\)  
25857.f3 25857h2 \([1, -1, 1, -827456, -279948054]\) \(17806161424897/668584449\) \(2352577358620372689\) \([2, 2]\) \(344064\) \(2.2938\)  
25857.f4 25857h1 \([1, -1, 1, -819851, -285520998]\) \(17319700013617/25857\) \(90984157428177\) \([2]\) \(172032\) \(1.9472\) \(\Gamma_0(N)\)-optimal
25857.f5 25857h3 \([1, -1, 1, 336109, -1005547188]\) \(1193377118543/124806800313\) \(-439163149931741597193\) \([2]\) \(688128\) \(2.6404\)  
25857.f6 25857h5 \([1, -1, 1, 5895364, 5427686580]\) \(6439735268725823/7345472585373\) \(-25846835831377783607853\) \([2]\) \(1376256\) \(2.9869\)