Show commands:
SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 110946a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
110946.i2 | 110946a1 | \([1, 1, 0, -51582360, 142572015936]\) | \(3195392484115617625/1508779008\) | \(7166857564632572928\) | \([2]\) | \(9676800\) | \(2.9551\) | \(\Gamma_0(N)\)-optimal |
110946.i3 | 110946a2 | \([1, 1, 0, -51313400, 144132683232]\) | \(-3145668549383265625/69470644988448\) | \(-329992805384632240807968\) | \([2]\) | \(19353600\) | \(3.3017\) | |
110946.i1 | 110946a3 | \([1, 1, 0, -61290135, 85159069653]\) | \(5360339745382407625/2442110888312832\) | \(11600281287567060617920512\) | \([2]\) | \(29030400\) | \(3.5044\) | |
110946.i4 | 110946a4 | \([1, 1, 0, 214124905, 639900043221]\) | \(228571521134288888375/169504391772536832\) | \(-805163530226852713011904512\) | \([2]\) | \(58060800\) | \(3.8510\) |
Rank
sage: E.rank()
The elliptic curves in class 110946a have rank \(1\).
Complex multiplication
The elliptic curves in class 110946a do not have complex multiplication.Modular form 110946.2.a.a
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.