Properties

Label 13872.bm
Number of curves $6$
Conductor $13872$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("bm1")
 
E.isogeny_class()
 

Elliptic curves in class 13872.bm

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
13872.bm1 13872bi5 \([0, 1, 0, -128288352, 559236806388]\) \(2361739090258884097/5202\) \(514308644610048\) \([2]\) \(884736\) \(2.9568\)  
13872.bm2 13872bi3 \([0, 1, 0, -8018112, 8735863860]\) \(576615941610337/27060804\) \(2675433569261469696\) \([2, 2]\) \(442368\) \(2.6103\)  
13872.bm3 13872bi6 \([0, 1, 0, -7601952, 9683543412]\) \(-491411892194497/125563633938\) \(-12414160396571511693312\) \([2]\) \(884736\) \(2.9568\)  
13872.bm4 13872bi2 \([0, 1, 0, -527232, 121351860]\) \(163936758817/30338064\) \(2999448015365799936\) \([2, 2]\) \(221184\) \(2.2637\)  
13872.bm5 13872bi1 \([0, 1, 0, -157312, -22325068]\) \(4354703137/352512\) \(34851974034751488\) \([2]\) \(110592\) \(1.9171\) \(\Gamma_0(N)\)-optimal
13872.bm6 13872bi4 \([0, 1, 0, 1044928, 708081972]\) \(1276229915423/2927177028\) \(-289402623953161961472\) \([4]\) \(442368\) \(2.6103\)  

Rank

sage: E.rank()
 

The elliptic curves in class 13872.bm have rank \(1\).

Complex multiplication

The elliptic curves in class 13872.bm do not have complex multiplication.

Modular form 13872.2.a.bm

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.