Properties

Label 630f
Number of curves $8$
Conductor $630$
CM no
Rank $0$
Graph

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

Elliptic curves in class 630f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
630.f7 630f1 \([1, -1, 0, -369, 1053]\) \(7633736209/3870720\) \(2821754880\) \([2]\) \(384\) \(0.50588\) \(\Gamma_0(N)\)-optimal
630.f5 630f2 \([1, -1, 0, -3249, -69795]\) \(5203798902289/57153600\) \(41664974400\) \([2, 2]\) \(768\) \(0.85245\)  
630.f4 630f3 \([1, -1, 0, -24129, 1448685]\) \(2131200347946769/2058000\) \(1500282000\) \([6]\) \(1152\) \(1.0552\)  
630.f2 630f4 \([1, -1, 0, -51849, -4531275]\) \(21145699168383889/2593080\) \(1890355320\) \([2]\) \(1536\) \(1.1990\)  
630.f6 630f5 \([1, -1, 0, -729, -177147]\) \(-58818484369/18600435000\) \(-13559717115000\) \([2]\) \(1536\) \(1.1990\)  
630.f3 630f6 \([1, -1, 0, -24309, 1426113]\) \(2179252305146449/66177562500\) \(48243443062500\) \([2, 6]\) \(2304\) \(1.4018\)  
630.f1 630f7 \([1, -1, 0, -58059, -3373137]\) \(29689921233686449/10380965400750\) \(7567723777146750\) \([6]\) \(4608\) \(1.7483\)  
630.f8 630f8 \([1, -1, 0, 6561, 4778595]\) \(42841933504271/13565917968750\) \(-9889554199218750\) \([6]\) \(4608\) \(1.7483\)  

Rank

sage: E.rank()
 

The elliptic curves in class 630f have rank \(0\).

Complex multiplication

The elliptic curves in class 630f do not have complex multiplication.

Modular form 630.2.a.f

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{4} + q^{5} + q^{7} - q^{8} - q^{10} + 2 q^{13} - q^{14} + q^{16} + 6 q^{17} - 4 q^{19} + O(q^{20})\) 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 Cremona numbering.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.