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SageMath
E = EllipticCurve("eh1")
E.isogeny_class()
Elliptic curves in class 25200.eh
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
25200.eh1 | 25200eg8 | \([0, 0, 0, -1264437075, -17305890272750]\) | \(4791901410190533590281/41160000\) | \(1920360960000000000\) | \([2]\) | \(5308416\) | \(3.5494\) | |
25200.eh2 | 25200eg6 | \([0, 0, 0, -79029075, -270391904750]\) | \(1169975873419524361/108425318400\) | \(5058691655270400000000\) | \([2, 2]\) | \(2654208\) | \(3.2029\) | |
25200.eh3 | 25200eg7 | \([0, 0, 0, -73269075, -311477984750]\) | \(-932348627918877961/358766164249920\) | \(-16738594159244267520000000\) | \([2]\) | \(5308416\) | \(3.5494\) | |
25200.eh4 | 25200eg5 | \([0, 0, 0, -15687075, -23494022750]\) | \(9150443179640281/184570312500\) | \(8611312500000000000000\) | \([2]\) | \(1769472\) | \(3.0001\) | |
25200.eh5 | 25200eg3 | \([0, 0, 0, -5301075, -3570272750]\) | \(353108405631241/86318776320\) | \(4027288827985920000000\) | \([2]\) | \(1327104\) | \(2.8563\) | |
25200.eh6 | 25200eg2 | \([0, 0, 0, -2079075, 605745250]\) | \(21302308926361/8930250000\) | \(416649744000000000000\) | \([2, 2]\) | \(884736\) | \(2.6536\) | |
25200.eh7 | 25200eg1 | \([0, 0, 0, -1791075, 922257250]\) | \(13619385906841/6048000\) | \(282175488000000000\) | \([2]\) | \(442368\) | \(2.3070\) | \(\Gamma_0(N)\)-optimal |
25200.eh8 | 25200eg4 | \([0, 0, 0, 6920925, 4448745250]\) | \(785793873833639/637994920500\) | \(-29766291010848000000000\) | \([2]\) | \(1769472\) | \(3.0001\) |
Rank
sage: E.rank()
The elliptic curves in class 25200.eh have rank \(0\).
Complex multiplication
The elliptic curves in class 25200.eh do not have complex multiplication.Modular form 25200.2.a.eh
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.