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Properties

Label 27075.f
Number of curves $8$
Conductor $27075$
CM no
Rank $2$
Graph

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

Elliptic curves in class 27075.f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
27075.f1 27075g8 \([1, 1, 1, -19494188, 33120687656]\) \(1114544804970241/405\) \(297712215703125\) \([2]\) \(663552\) \(2.5678\)  
27075.f2 27075g6 \([1, 1, 1, -1218563, 516972656]\) \(272223782641/164025\) \(120573447359765625\) \([2, 2]\) \(331776\) \(2.2212\)  
27075.f3 27075g7 \([1, 1, 1, -992938, 714620156]\) \(-147281603041/215233605\) \(-158216477625484453125\) \([2]\) \(663552\) \(2.5678\)  
27075.f4 27075g4 \([1, 1, 1, -722188, -236524594]\) \(56667352321/15\) \(11026378359375\) \([2]\) \(165888\) \(1.8747\)  
27075.f5 27075g3 \([1, 1, 1, -90438, 4803906]\) \(111284641/50625\) \(37214026962890625\) \([2, 2]\) \(165888\) \(1.8747\)  
27075.f6 27075g2 \([1, 1, 1, -45313, -3679594]\) \(13997521/225\) \(165395675390625\) \([2, 2]\) \(82944\) \(1.5281\)  
27075.f7 27075g1 \([1, 1, 1, -188, -159844]\) \(-1/15\) \(-11026378359375\) \([2]\) \(41472\) \(1.1815\) \(\Gamma_0(N)\)-optimal
27075.f8 27075g5 \([1, 1, 1, 315687, 36481656]\) \(4733169839/3515625\) \(-2584307427978515625\) \([2]\) \(331776\) \(2.2212\)  

Rank

sage: E.rank()
 

The elliptic curves in class 27075.f have rank \(2\).

Complex multiplication

The elliptic curves in class 27075.f do not have complex multiplication.

Modular form 27075.2.a.f

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} - q^{4} + q^{6} + 3 q^{8} + q^{9} - 4 q^{11} + q^{12} - 2 q^{13} - q^{16} - 2 q^{17} - q^{18} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 16 & 4 & 8 & 16 & 8 \\ 2 & 1 & 2 & 8 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 16 & 4 & 8 & 16 & 8 \\ 16 & 8 & 16 & 1 & 4 & 2 & 4 & 8 \\ 4 & 2 & 4 & 4 & 1 & 2 & 4 & 2 \\ 8 & 4 & 8 & 2 & 2 & 1 & 2 & 4 \\ 16 & 8 & 16 & 4 & 4 & 2 & 1 & 8 \\ 8 & 4 & 8 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.