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SageMath
E = EllipticCurve("n1")
E.isogeny_class()
Elliptic curves in class 433200.n
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
433200.n1 | 433200n8 | \([0, -1, 0, -770160408, -8226334076688]\) | \(16778985534208729/81000\) | \(243885847104000000000\) | \([2]\) | \(95551488\) | \(3.5343\) | |
433200.n2 | 433200n7 | \([0, -1, 0, -65488408, -27741052688]\) | \(10316097499609/5859375000\) | \(17642205375000000000000000\) | \([2]\) | \(95551488\) | \(3.5343\) | |
433200.n3 | 433200n6 | \([0, -1, 0, -48160408, -128382076688]\) | \(4102915888729/9000000\) | \(27098427456000000000000\) | \([2, 2]\) | \(47775744\) | \(3.1877\) | |
433200.n4 | 433200n4 | \([0, -1, 0, -41662408, 103518547312]\) | \(2656166199049/33750\) | \(101619102960000000000\) | \([2]\) | \(31850496\) | \(2.9849\) | \(\Gamma_0(N)\)-optimal* |
433200.n5 | 433200n5 | \([0, -1, 0, -9894408, -10314860688]\) | \(35578826569/5314410\) | \(16001350428493440000000\) | \([2]\) | \(31850496\) | \(2.9849\) | |
433200.n6 | 433200n2 | \([0, -1, 0, -2674408, 1525939312]\) | \(702595369/72900\) | \(219497262393600000000\) | \([2, 2]\) | \(15925248\) | \(2.6384\) | \(\Gamma_0(N)\)-optimal* |
433200.n7 | 433200n3 | \([0, -1, 0, -1952408, -3435644688]\) | \(-273359449/1536000\) | \(-4624798285824000000000\) | \([2]\) | \(23887872\) | \(2.8411\) | |
433200.n8 | 433200n1 | \([0, -1, 0, 213592, 116595312]\) | \(357911/2160\) | \(-6503622589440000000\) | \([2]\) | \(7962624\) | \(2.2918\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 433200.n have rank \(1\).
Complex multiplication
The elliptic curves in class 433200.n do not have complex multiplication.Modular form 433200.2.a.n
sage: E.q_eigenform(10)
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 4 & 2 & 12 & 3 & 6 & 4 & 12 \\ 4 & 1 & 2 & 3 & 12 & 6 & 4 & 12 \\ 2 & 2 & 1 & 6 & 6 & 3 & 2 & 6 \\ 12 & 3 & 6 & 1 & 4 & 2 & 12 & 4 \\ 3 & 12 & 6 & 4 & 1 & 2 & 12 & 4 \\ 6 & 6 & 3 & 2 & 2 & 1 & 6 & 2 \\ 4 & 4 & 2 & 12 & 12 & 6 & 1 & 3 \\ 12 & 12 & 6 & 4 & 4 & 2 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.