Show commands:
SageMath
E = EllipticCurve("bg1")
E.isogeny_class()
Elliptic curves in class 379050bg
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
379050.bg4 | 379050bg1 | \([1, 1, 0, -421715875, 3926242778125]\) | \(-11283450590382195961/2530373271552000\) | \(-1860056872172126208000000000\) | \([2]\) | \(232243200\) | \(3.9527\) | \(\Gamma_0(N)\)-optimal |
379050.bg3 | 379050bg2 | \([1, 1, 0, -7075667875, 229076016602125]\) | \(53294746224000958661881/1997017344000000\) | \(1467991255011876000000000000\) | \([2, 2]\) | \(464486400\) | \(4.2993\) | |
379050.bg1 | 379050bg3 | \([1, 1, 0, -113209667875, 14661283470602125]\) | \(218289391029690300712901881/306514992000\) | \(225316684974186750000000\) | \([2]\) | \(928972800\) | \(4.6459\) | |
379050.bg2 | 379050bg4 | \([1, 1, 0, -7404899875, 206587166378125]\) | \(61085713691774408830201/10268551781250000000\) | \(7548329142859773925781250000000\) | \([2]\) | \(928972800\) | \(4.6459\) |
Rank
sage: E.rank()
The elliptic curves in class 379050bg have rank \(1\).
Complex multiplication
The elliptic curves in class 379050bg do not have complex multiplication.Modular form 379050.2.a.bg
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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.