Show commands:
SageMath
E = EllipticCurve("qm1")
E.isogeny_class()
Elliptic curves in class 176400qm
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 176400.rt4 | 176400qm1 | \([0, 0, 0, -8107050, -8884686125]\) | \(2748251600896/2205\) | \(47278574201250000\) | \([2]\) | \(4718592\) | \(2.5050\) | \(\Gamma_0(N)\)-optimal |
| 176400.rt3 | 176400qm2 | \([0, 0, 0, -8162175, -8757733250]\) | \(175293437776/4862025\) | \(1667988097820100000000\) | \([2, 2]\) | \(9437184\) | \(2.8516\) | |
| 176400.rt2 | 176400qm3 | \([0, 0, 0, -18966675, 19301553250]\) | \(549871953124/200930625\) | \(275728644741690000000000\) | \([2, 2]\) | \(18874368\) | \(3.1982\) | |
| 176400.rt5 | 176400qm4 | \([0, 0, 0, 1760325, -28692035750]\) | \(439608956/259416045\) | \(-355985726476983120000000\) | \([2]\) | \(18874368\) | \(3.1982\) | |
| 176400.rt1 | 176400qm5 | \([0, 0, 0, -269013675, 1697867064250]\) | \(784478485879202/221484375\) | \(607867382587500000000000\) | \([2]\) | \(37748736\) | \(3.5447\) | |
| 176400.rt6 | 176400qm6 | \([0, 0, 0, 58208325, 136530378250]\) | \(7947184069438/7533176175\) | \(-20674921578859749600000000\) | \([2]\) | \(37748736\) | \(3.5447\) |
Rank
sage: E.rank()
The elliptic curves in class 176400qm have rank \(1\).
Complex multiplication
The elliptic curves in class 176400qm do not have complex multiplication.Modular form 176400.2.a.qm
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
