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Properties

Label 18928z
Number of curves $6$
Conductor $18928$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("z1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 18928z have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(7\)\(1 + T\)
\(13\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(3\) \( 1 - 2 T + 3 T^{2}\) 1.3.ac
\(5\) \( 1 + T + 5 T^{2}\) 1.5.b
\(11\) \( 1 + 4 T + 11 T^{2}\) 1.11.e
\(17\) \( 1 + 2 T + 17 T^{2}\) 1.17.c
\(19\) \( 1 + T + 19 T^{2}\) 1.19.b
\(23\) \( 1 - 7 T + 23 T^{2}\) 1.23.ah
\(29\) \( 1 + 5 T + 29 T^{2}\) 1.29.f
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 18928z do not have complex multiplication.

Modular form 18928.2.a.z

Copy content sage:E.q_eigenform(10)
 
\(q + 2 q^{3} + q^{7} + q^{9} + 6 q^{17} + 2 q^{19} + O(q^{20})\) 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 Cremona numbering.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 6 & 9 & 18 \\ 2 & 1 & 6 & 3 & 18 & 9 \\ 3 & 6 & 1 & 2 & 3 & 6 \\ 6 & 3 & 2 & 1 & 6 & 3 \\ 9 & 18 & 3 & 6 & 1 & 2 \\ 18 & 9 & 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 18928z

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
18928.bb5 18928z1 \([0, -1, 0, -1408, -40704]\) \(-15625/28\) \(-553577070592\) \([2]\) \(17280\) \(0.94354\) \(\Gamma_0(N)\)-optimal
18928.bb4 18928z2 \([0, -1, 0, -28448, -1836160]\) \(128787625/98\) \(1937519747072\) \([2]\) \(34560\) \(1.2901\)  
18928.bb6 18928z3 \([0, -1, 0, 12112, 857024]\) \(9938375/21952\) \(-434004423344128\) \([2]\) \(51840\) \(1.4928\)  
18928.bb3 18928z4 \([0, -1, 0, -96048, 9423296]\) \(4956477625/941192\) \(18607939650879488\) \([2]\) \(103680\) \(1.8394\)  
18928.bb2 18928z5 \([0, -1, 0, -461088, 121011968]\) \(-548347731625/1835008\) \(-36279226898317312\) \([2]\) \(155520\) \(2.0421\)  
18928.bb1 18928z6 \([0, -1, 0, -7383328, 7724400384]\) \(2251439055699625/25088\) \(496005055250432\) \([2]\) \(311040\) \(2.3887\)