Properties

Label 672.b
Number of curves $4$
Conductor $672$
CM no
Rank $1$
Graph

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

Elliptic curves in class 672.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
672.b1 672e3 \([0, -1, 0, -224, 1368]\) \(2438569736/21\) \(10752\) \([2]\) \(128\) \(-0.058633\)  
672.b2 672e2 \([0, -1, 0, -49, -95]\) \(3241792/567\) \(2322432\) \([2]\) \(128\) \(-0.058633\)  
672.b3 672e1 \([0, -1, 0, -14, 24]\) \(5088448/441\) \(28224\) \([2, 2]\) \(64\) \(-0.40521\) \(\Gamma_0(N)\)-optimal
672.b4 672e4 \([0, -1, 0, 16, 84]\) \(830584/7203\) \(-3687936\) \([4]\) \(128\) \(-0.058633\)  

Rank

sage: E.rank()
 

The elliptic curves in class 672.b have rank \(1\).

Complex multiplication

The elliptic curves in class 672.b do not have complex multiplication.

Modular form 672.2.a.b

sage: E.q_eigenform(10)
 
\(q - q^{3} - 2 q^{5} + q^{7} + q^{9} + 4 q^{11} - 6 q^{13} + 2 q^{15} - 2 q^{17} - 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.