Properties

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

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Show commands: SageMath
E = EllipticCurve("p1")
 
E.isogeny_class()
 

Elliptic curves in class 27378.p

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
27378.p1 27378s3 \([1, -1, 1, -182045, 29941669]\) \(-189613868625/128\) \(-450399201408\) \([]\) \(96768\) \(1.5518\)  
27378.p2 27378s4 \([1, -1, 1, -144020, 42769783]\) \(-1159088625/2097152\) \(-597726581785362432\) \([]\) \(290304\) \(2.1012\)  
27378.p3 27378s2 \([1, -1, 1, -7130, -241055]\) \(-140625/8\) \(-2280145957128\) \([]\) \(41472\) \(1.1282\)  
27378.p4 27378s1 \([1, -1, 1, 475, -737]\) \(3375/2\) \(-7037487522\) \([]\) \(13824\) \(0.57889\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 27378.2.a.p

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.