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SageMath
E = EllipticCurve("he1")
E.isogeny_class()
Elliptic curves in class 129360.he
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 129360.he1 | 129360ia7 | \([0, 1, 0, -4174016800, -103797101039500]\) | \(16689299266861680229173649/2396798250\) | \(1154993837319168000\) | \([2]\) | \(47775744\) | \(3.7885\) | |
| 129360.he2 | 129360ia8 | \([0, 1, 0, -267736800, -1532093215500]\) | \(4404531606962679693649/444872222400201750\) | \(214379610493588830968832000\) | \([4]\) | \(47775744\) | \(3.7885\) | |
| 129360.he3 | 129360ia6 | \([0, 1, 0, -260876800, -1621885127500]\) | \(4074571110566294433649/48828650062500\) | \(23530053022527744000000\) | \([2, 2]\) | \(23887872\) | \(3.4420\) | |
| 129360.he4 | 129360ia5 | \([0, 1, 0, -58808640, 173221101108]\) | \(46676570542430835889/106752955783320\) | \(51443214315322632929280\) | \([4]\) | \(15925248\) | \(3.2392\) | |
| 129360.he5 | 129360ia4 | \([0, 1, 0, -51595840, -142022288332]\) | \(31522423139920199089/164434491947880\) | \(79239387312849445355520\) | \([2]\) | \(15925248\) | \(3.2392\) | |
| 129360.he6 | 129360ia3 | \([0, 1, 0, -15876800, -26739127500]\) | \(-918468938249433649/109183593750000\) | \(-52614515184000000000000\) | \([2]\) | \(11943936\) | \(3.0954\) | |
| 129360.he7 | 129360ia2 | \([0, 1, 0, -5026240, 536571188]\) | \(29141055407581489/16604321025600\) | \(8001461306739975782400\) | \([2, 2]\) | \(7962624\) | \(2.8926\) | |
| 129360.he8 | 129360ia1 | \([0, 1, 0, 1245760, 67425588]\) | \(443688652450511/260789760000\) | \(-125672056726487040000\) | \([2]\) | \(3981312\) | \(2.5461\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 129360.he have rank \(0\).
Complex multiplication
The elliptic curves in class 129360.he do not have complex multiplication.Modular form 129360.2.a.he
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.
