Show commands:
SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 7623.f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 7623.f1 | 7623i5 | \([1, -1, 1, -4921214, -4200771594]\) | \(10206027697760497/5557167\) | \(7176903178883823\) | \([2]\) | \(153600\) | \(2.3710\) | |
| 7623.f2 | 7623i3 | \([1, -1, 1, -309299, -64806222]\) | \(2533811507137/58110129\) | \(75047370277958001\) | \([2, 2]\) | \(76800\) | \(2.0245\) | |
| 7623.f3 | 7623i2 | \([1, -1, 1, -42494, 1895028]\) | \(6570725617/2614689\) | \(3376787092396641\) | \([2, 2]\) | \(38400\) | \(1.6779\) | |
| 7623.f4 | 7623i1 | \([1, -1, 1, -37049, 2753160]\) | \(4354703137/1617\) | \(2088303705873\) | \([4]\) | \(19200\) | \(1.3313\) | \(\Gamma_0(N)\)-optimal |
| 7623.f5 | 7623i6 | \([1, -1, 1, 33736, -200922510]\) | \(3288008303/13504609503\) | \(-17440770606977509407\) | \([2]\) | \(153600\) | \(2.3710\) | |
| 7623.f6 | 7623i4 | \([1, -1, 1, 137191, 13610490]\) | \(221115865823/190238433\) | \(-245686842692252577\) | \([2]\) | \(76800\) | \(2.0245\) |
Rank
sage: E.rank()
The elliptic curves in class 7623.f have rank \(1\).
Complex multiplication
The elliptic curves in class 7623.f do not have complex multiplication.Modular form 7623.2.a.f
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 8 & 4 & 2 & 1 & 8 & 4 \\ 4 & 2 & 4 & 8 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
