Properties

Label 270480fh
Number of curves $6$
Conductor $270480$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("fh1")
 
E.isogeny_class()
 

Elliptic curves in class 270480fh

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
270480.fh4 270480fh1 \([0, 1, 0, -595856, 176807700]\) \(48551226272641/9273600\) \(4468857923174400\) \([2]\) \(2359296\) \(2.0038\) \(\Gamma_0(N)\)-optimal
270480.fh3 270480fh2 \([0, 1, 0, -658576, 137243924]\) \(65553197996161/20996010000\) \(10117773641687040000\) \([2, 2]\) \(4718592\) \(2.3504\)  
270480.fh6 270480fh3 \([0, 1, 0, 1865904, 938008980]\) \(1490881681033919/1650501562500\) \(-795360699705600000000\) \([2]\) \(9437184\) \(2.6970\)  
270480.fh2 270480fh4 \([0, 1, 0, -4186576, -3194599276]\) \(16840406336564161/604708416900\) \(291403122851299737600\) \([2, 2]\) \(9437184\) \(2.6970\)  
270480.fh5 270480fh5 \([0, 1, 0, 1575824, -11305753516]\) \(898045580910239/115117148363070\) \(-55473837620292904673280\) \([2]\) \(18874368\) \(3.0435\)  
270480.fh1 270480fh6 \([0, 1, 0, -66396976, -208264961836]\) \(67176973097223766561/91487391870\) \(44086887080401428480\) \([2]\) \(18874368\) \(3.0435\)  

Rank

sage: E.rank()
 

The elliptic curves in class 270480fh have rank \(1\).

Complex multiplication

The elliptic curves in class 270480fh do not have complex multiplication.

Modular form 270480.2.a.fh

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

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

Isogeny graph

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

The vertices are labelled with Cremona labels.