Show commands:
SageMath
E = EllipticCurve("g1")
E.isogeny_class()
Elliptic curves in class 6720g
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
6720.p3 | 6720g1 | \([0, -1, 0, -161, 801]\) | \(1771561/105\) | \(27525120\) | \([2]\) | \(2048\) | \(0.18061\) | \(\Gamma_0(N)\)-optimal |
6720.p2 | 6720g2 | \([0, -1, 0, -481, -2975]\) | \(47045881/11025\) | \(2890137600\) | \([2, 2]\) | \(4096\) | \(0.52719\) | |
6720.p1 | 6720g3 | \([0, -1, 0, -7201, -232799]\) | \(157551496201/13125\) | \(3440640000\) | \([2]\) | \(8192\) | \(0.87376\) | |
6720.p4 | 6720g4 | \([0, -1, 0, 1119, -19935]\) | \(590589719/972405\) | \(-254910136320\) | \([2]\) | \(8192\) | \(0.87376\) |
Rank
sage: E.rank()
The elliptic curves in class 6720g have rank \(0\).
Complex multiplication
The elliptic curves in class 6720g do not have complex multiplication.Modular form 6720.2.a.g
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.