Minimal Weierstrass equation
\(y^2+xy+y=x^3+x^2-73676513x-350955487969\)
Mordell-Weil group structure
$\Z\times \Z/{2}\Z$
Infinite order Mordell-Weil generator and height
| $P$ | = |
\(\left(\frac{16425975}{961}, \frac{54546228592}{29791}\right)\)
|
| $\hat{h}(P)$ | ≈ | $11.970484516500050856784562616$ |
Torsion generators
\( \left(\frac{41475}{4}, -\frac{41479}{8}\right) \)
Integral points
None
Invariants
|
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
|
|||
| Conductor: | \( 364650 \) | = | $2 \cdot 3 \cdot 5^{2} \cdot 11 \cdot 13 \cdot 17$ |
|
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
|
|||
| Discriminant: | $-27604316791839408192000000 $ | = | $-1 \cdot 2^{12} \cdot 3^{2} \cdot 5^{6} \cdot 11^{2} \cdot 13^{12} \cdot 17 $ |
|
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
|
|||
| j-invariant: | \( -\frac{2830680648734534916567625}{1766676274677722124288} \) | = | $-1 \cdot 2^{-12} \cdot 3^{-2} \cdot 5^{3} \cdot 7^{3} \cdot 11^{-2} \cdot 13^{-12} \cdot 17^{-1} \cdot 4041683^{3}$ |
| Endomorphism ring: | $\Z$ | ||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
| Sato-Tate group: | $\mathrm{SU}(2)$ | ||
| Faltings height: | $3.5833583660161991063042564980\dots$ | ||
| Stable Faltings height: | $2.7786394097991489190038768314\dots$ | ||
BSD invariants
sage: E.rank()
magma: Rank(E);
| |||
| Analytic rank: | $1$ | ||
sage: E.regulator()
magma: Regulator(E);
| |||
| Regulator: | $11.970484516500050856784562616\dots$ | ||
sage: E.period_lattice().omega()
gp: E.omega[1]
magma: RealPeriod(E);
| |||
| Real period: | $0.025056391275898407157546935936\dots$ | ||
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
magma: TamagawaNumbers(E);
| |||
| Tamagawa product: | $ 192 $ = $ ( 2^{2} \cdot 3 )\cdot2\cdot2\cdot2\cdot2\cdot1 $ | ||
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
| |||
| Torsion order: | $2$ | ||
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
| |||
| Analytic order of Ш: | $1$ (exact) | ||
sage: r = E.rank();
gp: ar = ellanalyticrank(E);
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
| |||
| Special value: | $ L'(E,1) $ ≈ $ 14.396982902760424166108227570 $ | ||
Modular invariants
Modular form 364650.2.a.ey
For more coefficients, see the Downloads section to the right.
|
sage: E.modular_degree()
magma: ModularDegree(E);
|
|||
| Modular degree: | 131383296 | ||
| $ \Gamma_0(N) $-optimal: | no | ||
| Manin constant: | 1 | ||
Local data
This elliptic curve is not semistable. There are 6 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
|---|---|---|---|---|---|---|---|
| $2$ | $12$ | $I_{12}$ | Split multiplicative | -1 | 1 | 12 | 12 |
| $3$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
| $5$ | $2$ | $I_0^{*}$ | Additive | 1 | 2 | 6 | 0 |
| $11$ | $2$ | $I_{2}$ | Split multiplicative | -1 | 1 | 2 | 2 |
| $13$ | $2$ | $I_{12}$ | Non-split multiplicative | 1 | 1 | 12 | 12 |
| $17$ | $1$ | $I_{1}$ | Non-split multiplicative | 1 | 1 | 1 | 1 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$ except those listed in the table below.
| prime $\ell$ | mod-$\ell$ image | $\ell$-adic image |
|---|---|---|
| $2$ | 2B | 2.3.0.1 |
| $3$ | 3B | 3.4.0.1 |
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.
No Iwasawa invariant data is available for this curve.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 364650.ey
consists of 4 curves linked by isogenies of
degrees dividing 6.
Growth of torsion in number fields
The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-17}) \) | \(\Z/2\Z \times \Z/2\Z\) | Not in database |
| $2$ | \(\Q(\sqrt{-15}) \) | \(\Z/6\Z\) | Not in database |
| $4$ | 4.2.1851300.1 | \(\Z/4\Z\) | Not in database |
| $4$ | \(\Q(\sqrt{-15}, \sqrt{-17})\) | \(\Z/2\Z \times \Z/6\Z\) | Not in database |
| $6$ | 6.2.9025868177011125.3 | \(\Z/6\Z\) | Not in database |
| $8$ | Deg 8 | \(\Z/2\Z \times \Z/4\Z\) | Not in database |
| $8$ | 8.0.15847889254560000.8 | \(\Z/2\Z \times \Z/4\Z\) | Not in database |
| $8$ | 8.0.3427311690000.7 | \(\Z/12\Z\) | Not in database |
| $12$ | Deg 12 | \(\Z/3\Z \times \Z/6\Z\) | Not in database |
| $12$ | Deg 12 | \(\Z/2\Z \times \Z/6\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/8\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/2\Z \times \Z/12\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/2\Z \times \Z/12\Z\) | Not in database |
| $18$ | 18.0.172147992785701740832251062094080728425091177176000000000.1 | \(\Z/18\Z\) | Not in database |
We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.