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SageMath
E = EllipticCurve("s1")
E.isogeny_class()
Elliptic curves in class 48510.s
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 48510.s1 | 48510ba7 | \([1, -1, 0, -2347884450, -43788228058814]\) | \(16689299266861680229173649/2396798250\) | \(205564088722088250\) | \([2]\) | \(15925248\) | \(3.6447\) | |
| 48510.s2 | 48510ba8 | \([1, -1, 0, -150601950, -646276524314]\) | \(4404531606962679693649/444872222400201750\) | \(38154964855914613714911750\) | \([2]\) | \(15925248\) | \(3.6447\) | |
| 48510.s3 | 48510ba6 | \([1, -1, 0, -146743200, -684159416564]\) | \(4074571110566294433649/48828650062500\) | \(4187843909527032562500\) | \([2, 2]\) | \(7962624\) | \(3.2981\) | |
| 48510.s4 | 48510ba5 | \([1, -1, 0, -33079860, 73094191960]\) | \(46676570542430835889/106752955783320\) | \(9155786922819872901720\) | \([2]\) | \(5308416\) | \(3.0954\) | |
| 48510.s5 | 48510ba4 | \([1, -1, 0, -29022660, -59901141560]\) | \(31522423139920199089/164434491947880\) | \(14102908532975401773480\) | \([2]\) | \(5308416\) | \(3.0954\) | |
| 48510.s6 | 48510ba3 | \([1, -1, 0, -8930700, -11276104064]\) | \(-918468938249433649/109183593750000\) | \(-9364253312777343750000\) | \([2]\) | \(3981312\) | \(2.9515\) | |
| 48510.s7 | 48510ba2 | \([1, -1, 0, -2827260, 227779600]\) | \(29141055407581489/16604321025600\) | \(1424088206204453697600\) | \([2, 2]\) | \(2654208\) | \(2.7488\) | |
| 48510.s8 | 48510ba1 | \([1, -1, 0, 700740, 28094800]\) | \(443688652450511/260789760000\) | \(-22366926111720960000\) | \([2]\) | \(1327104\) | \(2.4022\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 48510.s have rank \(0\).
Complex multiplication
The elliptic curves in class 48510.s do not have complex multiplication.Modular form 48510.2.a.s
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 4 & 2 & 12 & 3 & 4 & 6 & 12 \\ 4 & 1 & 2 & 3 & 12 & 4 & 6 & 12 \\ 2 & 2 & 1 & 6 & 6 & 2 & 3 & 6 \\ 12 & 3 & 6 & 1 & 4 & 12 & 2 & 4 \\ 3 & 12 & 6 & 4 & 1 & 12 & 2 & 4 \\ 4 & 4 & 2 & 12 & 12 & 1 & 6 & 3 \\ 6 & 6 & 3 & 2 & 2 & 6 & 1 & 2 \\ 12 & 12 & 6 & 4 & 4 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
