Show commands:
SageMath
E = EllipticCurve("el1")
E.isogeny_class()
Elliptic curves in class 117810el
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 117810.ei7 | 117810el1 | \([1, -1, 1, -8510882, 80099489]\) | \(93523304529581769096409/54118679989886265600\) | \(39452517712627087622400\) | \([4]\) | \(10616832\) | \(3.0252\) | \(\Gamma_0(N)\)-optimal |
| 117810.ei5 | 117810el2 | \([1, -1, 1, -93218162, -345322305439]\) | \(122884692280581205924284889/439106354595306090000\) | \(320108532499978139610000\) | \([2, 2]\) | \(21233664\) | \(3.3718\) | |
| 117810.ei4 | 117810el3 | \([1, -1, 1, -475589417, 3992151134441]\) | \(16318969429297971769640983369/102045248126976000000\) | \(74390985884565504000000\) | \([12]\) | \(31850496\) | \(3.5746\) | |
| 117810.ei6 | 117810el4 | \([1, -1, 1, -51543662, -655680641839]\) | \(-20774088968758822168212889/242753662862303369030100\) | \(-176967420226619156022942900\) | \([2]\) | \(42467328\) | \(3.7184\) | |
| 117810.ei2 | 117810el5 | \([1, -1, 1, -1490209142, -22141734371791]\) | \(502039459750388822744052370969/6444603154532812500\) | \(4698115699654420312500\) | \([2]\) | \(42467328\) | \(3.7184\) | |
| 117810.ei3 | 117810el6 | \([1, -1, 1, -484621097, 3832648052969]\) | \(17266453047612484705388895049/1288004819409000000000000\) | \(938955513349161000000000000\) | \([2, 6]\) | \(63700992\) | \(3.9211\) | |
| 117810.ei8 | 117810el7 | \([1, -1, 1, 460378903, 16965502052969]\) | \(14802750729576629005731104951/179133615680899546821000000\) | \(-130588405831375769632509000000\) | \([6]\) | \(127401984\) | \(4.2677\) | |
| 117810.ei1 | 117810el8 | \([1, -1, 1, -1574127977, -19508511541399]\) | \(591720065532918583239955136329/116891407012939453125000000\) | \(85213835712432861328125000000\) | \([6]\) | \(127401984\) | \(4.2677\) |
Rank
sage: E.rank()
The elliptic curves in class 117810el have rank \(0\).
Complex multiplication
The elliptic curves in class 117810el do not have complex multiplication.Modular form 117810.2.a.el
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
