Properties

Label 418950b
Number of curves $6$
Conductor $418950$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("b1")
 
E.isogeny_class()
 

Elliptic curves in class 418950b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
418950.b6 418950b1 \([1, -1, 0, -86000742, 306993890916]\) \(52492168638015625/293197968\) \(392913318756908250000\) \([2]\) \(63700992\) \(3.1440\) \(\Gamma_0(N)\)-optimal
418950.b5 418950b2 \([1, -1, 0, -87544242, 295403749416]\) \(55369510069623625/3916046302812\) \(5247876578884009878937500\) \([2]\) \(127401984\) \(3.4906\)  
418950.b4 418950b3 \([1, -1, 0, -123210117, 16211555541]\) \(154357248921765625/89242711068672\) \(119593768060683782208000000\) \([2]\) \(191102976\) \(3.6933\)  
418950.b3 418950b4 \([1, -1, 0, -1333314117, -18676264932459]\) \(195607431345044517625/752875610010048\) \(1008925322907352937247000000\) \([2]\) \(382205952\) \(4.0399\)  
418950.b2 418950b5 \([1, -1, 0, -6743667492, -213150352753584]\) \(25309080274342544331625/191933498523648\) \(257209244661445558272000000\) \([2]\) \(573308928\) \(4.2426\)  
418950.b1 418950b6 \([1, -1, 0, -107898483492, -13641755641201584]\) \(103665426767620308239307625/5961940992\) \(7989571133042688000000\) \([2]\) \(1146617856\) \(4.5892\)  

Rank

sage: E.rank()
 

The elliptic curves in class 418950b have rank \(1\).

Complex multiplication

The elliptic curves in class 418950b do not have complex multiplication.

Modular form 418950.2.a.b

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{4} - q^{8} - 6 q^{11} - 4 q^{13} + q^{16} - q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 6 & 9 & 18 \\ 2 & 1 & 6 & 3 & 18 & 9 \\ 3 & 6 & 1 & 2 & 3 & 6 \\ 6 & 3 & 2 & 1 & 6 & 3 \\ 9 & 18 & 3 & 6 & 1 & 2 \\ 18 & 9 & 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.