Properties

Label 6720.bb
Number of curves $8$
Conductor $6720$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("bb1")
 
E.isogeny_class()
 

Elliptic curves in class 6720.bb

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
6720.bb1 6720l7 \([0, -1, 0, -412865, 64377537]\) \(29689921233686449/10380965400750\) \(2721307794014208000\) \([4]\) \(110592\) \(2.2387\)  
6720.bb2 6720l4 \([0, -1, 0, -368705, 86295105]\) \(21145699168383889/2593080\) \(679760363520\) \([4]\) \(36864\) \(1.6894\)  
6720.bb3 6720l6 \([0, -1, 0, -172865, -26870463]\) \(2179252305146449/66177562500\) \(17348050944000000\) \([2, 2]\) \(55296\) \(1.8922\)  
6720.bb4 6720l3 \([0, -1, 0, -171585, -27299775]\) \(2131200347946769/2058000\) \(539492352000\) \([2]\) \(27648\) \(1.5456\)  
6720.bb5 6720l2 \([0, -1, 0, -23105, 1346625]\) \(5203798902289/57153600\) \(14982473318400\) \([2, 2]\) \(18432\) \(1.3429\)  
6720.bb6 6720l5 \([0, -1, 0, -5185, 3364417]\) \(-58818484369/18600435000\) \(-4875992432640000\) \([2]\) \(36864\) \(1.6894\)  
6720.bb7 6720l1 \([0, -1, 0, -2625, -17343]\) \(7633736209/3870720\) \(1014686023680\) \([2]\) \(9216\) \(0.99629\) \(\Gamma_0(N)\)-optimal
6720.bb8 6720l8 \([0, -1, 0, 46655, -90662975]\) \(42841933504271/13565917968750\) \(-3556224000000000000\) \([2]\) \(110592\) \(2.2387\)  

Rank

sage: E.rank()
 

The elliptic curves in class 6720.bb have rank \(1\).

Complex multiplication

The elliptic curves in class 6720.bb do not have complex multiplication.

Modular form 6720.2.a.bb

sage: E.q_eigenform(10)
 
\(q - q^{3} + q^{5} + q^{7} + q^{9} - 2 q^{13} - q^{15} - 6 q^{17} + 4 q^{19} + O(q^{20})\) 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 LMFDB numbering.

\(\left(\begin{array}{rrrrrrrr} 1 & 3 & 2 & 4 & 6 & 12 & 12 & 4 \\ 3 & 1 & 6 & 12 & 2 & 4 & 4 & 12 \\ 2 & 6 & 1 & 2 & 3 & 6 & 6 & 2 \\ 4 & 12 & 2 & 1 & 6 & 12 & 3 & 4 \\ 6 & 2 & 3 & 6 & 1 & 2 & 2 & 6 \\ 12 & 4 & 6 & 12 & 2 & 1 & 4 & 3 \\ 12 & 4 & 6 & 3 & 2 & 4 & 1 & 12 \\ 4 & 12 & 2 & 4 & 6 & 3 & 12 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.