Properties

Label 68450.u
Number of curves $2$
Conductor $68450$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("u1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 68450.u have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(5\)\(1\)
\(37\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(3\) \( 1 - 2 T + 3 T^{2}\) 1.3.ac
\(7\) \( 1 - 4 T + 7 T^{2}\) 1.7.ae
\(11\) \( 1 + 11 T^{2}\) 1.11.a
\(13\) \( 1 - 2 T + 13 T^{2}\) 1.13.ac
\(17\) \( 1 + 17 T^{2}\) 1.17.a
\(19\) \( 1 + 5 T + 19 T^{2}\) 1.19.f
\(23\) \( 1 - 3 T + 23 T^{2}\) 1.23.ad
\(29\) \( 1 + 6 T + 29 T^{2}\) 1.29.g
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 68450.u do not have complex multiplication.

Modular form 68450.2.a.u

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} + 2 q^{3} + q^{4} - 2 q^{6} + 4 q^{7} - q^{8} + q^{9} + 2 q^{12} + 2 q^{13} - 4 q^{14} + q^{16} - q^{18} - 5 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 68450.u

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
68450.u1 68450f1 \([1, 1, 0, -16274700, 60383474000]\) \(-19026212425/51868672\) \(-1299617397950770000000000\) \([]\) \(12312000\) \(3.3144\) \(\Gamma_0(N)\)-optimal
68450.u2 68450f2 \([1, 1, 0, 142015925, -1371038647875]\) \(12642252501575/39728447488\) \(-995432879966126080000000000\) \([]\) \(36936000\) \(3.8637\)