Properties

Label 27378.f
Number of curves $4$
Conductor $27378$
CM no
Rank $1$
Graph

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

Elliptic curves in class 27378.f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
27378.f1 27378b4 \([1, -1, 0, -1638402, -806786668]\) \(-189613868625/128\) \(-328341017826432\) \([]\) \(290304\) \(2.1012\)  
27378.f2 27378b3 \([1, -1, 0, -16002, -1578732]\) \(-1159088625/2097152\) \(-819926723985408\) \([]\) \(96768\) \(1.5518\)  
27378.f3 27378b1 \([1, -1, 0, -792, 9192]\) \(-140625/8\) \(-3127772232\) \([]\) \(13824\) \(0.57889\) \(\Gamma_0(N)\)-optimal
27378.f4 27378b2 \([1, -1, 0, 4278, 15614]\) \(3375/2\) \(-5130328403538\) \([]\) \(41472\) \(1.1282\)  

Rank

sage: E.rank()
 

The elliptic curves in class 27378.f have rank \(1\).

Complex multiplication

The elliptic curves in class 27378.f do not have complex multiplication.

Modular form 27378.2.a.f

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{4} - 2 q^{7} - q^{8} + 3 q^{11} + 2 q^{14} + q^{16} - 3 q^{17} + q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 3 & 21 & 7 \\ 3 & 1 & 7 & 21 \\ 21 & 7 & 1 & 3 \\ 7 & 21 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.