Properties

Label 76440.b
Number of curves $6$
Conductor $76440$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("b1")
 
E.isogeny_class()
 

Elliptic curves in class 76440.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
76440.b1 76440bt6 \([0, -1, 0, -5339056, 4749294796]\) \(69855246474511682/14613770535\) \(3521117162848696320\) \([2]\) \(2359296\) \(2.5550\)  
76440.b2 76440bt4 \([0, -1, 0, -2379456, -1411953444]\) \(12367124507424964/14926275\) \(1798206799334400\) \([2]\) \(1179648\) \(2.2084\)  
76440.b3 76440bt3 \([0, -1, 0, -370456, 56948956]\) \(46670944188964/15429366225\) \(1858815495173145600\) \([2, 2]\) \(1179648\) \(2.2084\)  
76440.b4 76440bt2 \([0, -1, 0, -149956, -21637244]\) \(12381975627856/419225625\) \(12626297742240000\) \([2, 2]\) \(589824\) \(1.8618\)  
76440.b5 76440bt1 \([0, -1, 0, 3169, -1179744]\) \(1869154304/319921875\) \(-602215818750000\) \([2]\) \(294912\) \(1.5152\) \(\Gamma_0(N)\)-optimal
76440.b6 76440bt5 \([0, -1, 0, 1070144, 390591916]\) \(562511980386718/599562079935\) \(-144461576483374725120\) \([2]\) \(2359296\) \(2.5550\)  

Rank

sage: E.rank()
 

The elliptic curves in class 76440.b have rank \(1\).

Complex multiplication

The elliptic curves in class 76440.b do not have complex multiplication.

Modular form 76440.2.a.b

sage: E.q_eigenform(10)
 
\(q - q^{3} - q^{5} + q^{9} - 4 q^{11} - q^{13} + q^{15} - 2 q^{17} - 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.