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SageMath
E = EllipticCurve("dz1")
E.isogeny_class()
Elliptic curves in class 265200.dz
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
265200.dz1 | 265200dz7 | \([0, 1, 0, -2636400002408, 1647649523114107188]\) | \(31664865542564944883878115208137569/103216295812500\) | \(6605842932000000000000\) | \([2]\) | \(1911029760\) | \(5.2721\) | |
265200.dz2 | 265200dz6 | \([0, 1, 0, -164775002408, 25744481864107188]\) | \(7730680381889320597382223137569/441370202660156250000\) | \(28247692970250000000000000000\) | \([2, 2]\) | \(955514880\) | \(4.9255\) | |
265200.dz3 | 265200dz8 | \([0, 1, 0, -164476210408, 25842499384539188]\) | \(-7688701694683937879808871873249/58423707246780395507812500\) | \(-3739117263793945312500000000000000\) | \([2]\) | \(1911029760\) | \(5.2721\) | |
265200.dz4 | 265200dz4 | \([0, 1, 0, -32549306408, 2259971694187188]\) | \(59589391972023341137821784609/8834417507562311995200\) | \(565402720483987967692800000000\) | \([2]\) | \(637009920\) | \(4.7228\) | |
265200.dz5 | 265200dz3 | \([0, 1, 0, -10317114408, 400722685291188]\) | \(1897660325010178513043539489/14258428094958372000000\) | \(912539398077335808000000000000\) | \([2]\) | \(477757440\) | \(4.5789\) | |
265200.dz6 | 265200dz2 | \([0, 1, 0, -2222906408, 28373223787188]\) | \(18980483520595353274840609/5549773448629762560000\) | \(355185500712304803840000000000\) | \([2, 2]\) | \(318504960\) | \(4.3762\) | |
265200.dz7 | 265200dz1 | \([0, 1, 0, -838458408, -8998565524812]\) | \(1018563973439611524445729/42904970360310988800\) | \(2745918103059903283200000000\) | \([2]\) | \(159252480\) | \(4.0296\) | \(\Gamma_0(N)\)-optimal |
265200.dz8 | 265200dz5 | \([0, 1, 0, 5952325592, 188591420523188]\) | \(364421318680576777174674911/450962301637624725000000\) | \(-28861587304807982400000000000000\) | \([2]\) | \(637009920\) | \(4.7228\) |
Rank
sage: E.rank()
The elliptic curves in class 265200.dz have rank \(1\).
Complex multiplication
The elliptic curves in class 265200.dz do not have complex multiplication.Modular form 265200.2.a.dz
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 3 & 4 & 6 & 12 & 12 \\ 2 & 1 & 2 & 6 & 2 & 3 & 6 & 6 \\ 4 & 2 & 1 & 12 & 4 & 6 & 12 & 3 \\ 3 & 6 & 12 & 1 & 12 & 2 & 4 & 4 \\ 4 & 2 & 4 & 12 & 1 & 6 & 3 & 12 \\ 6 & 3 & 6 & 2 & 6 & 1 & 2 & 2 \\ 12 & 6 & 12 & 4 & 3 & 2 & 1 & 4 \\ 12 & 6 & 3 & 4 & 12 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.