Minimal Weierstrass equation
\(y^2+xy=x^3+x^2-8352x-272484\)
Mordell-Weil group structure
$\Z^2 \times \Z/{2}\Z$
Infinite order Mordell-Weil generators and heights
| $P$ | = |
\(\left(-58, 174\right)\)
|
\(\left(117, 549\right)\)
|
| $\hat{h}(P)$ | ≈ | $0.47926636601742276197921688321$ | $2.1291088932621181462982394504$ |
Torsion generators
\( \left(-\frac{261}{4}, \frac{261}{8}\right) \)
Integral points
\( \left(-58, 174\right) \), \( \left(-58, -116\right) \), \( \left(-53, 184\right) \), \( \left(-53, -131\right) \), \( \left(-45, 144\right) \), \( \left(-45, -99\right) \), \( \left(116, 522\right) \), \( \left(116, -638\right) \), \( \left(117, 549\right) \), \( \left(117, -666\right) \), \( \left(145, 1189\right) \), \( \left(145, -1334\right) \), \( \left(290, 4524\right) \), \( \left(290, -4814\right) \), \( \left(522, 11484\right) \), \( \left(522, -12006\right) \), \( \left(3770, 229564\right) \), \( \left(3770, -233334\right) \), \( \left(1409922, 1673439084\right) \), \( \left(1409922, -1674849006\right) \)
Invariants
|
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
|
|||
| Conductor: | \( 11310 \) | = | $2 \cdot 3 \cdot 5 \cdot 13 \cdot 29$ |
|
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
|
|||
| Discriminant: | $6032611833300 $ | = | $2^{2} \cdot 3^{8} \cdot 5^{2} \cdot 13 \cdot 29^{4} $ |
|
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
|
|||
| j-invariant: | \( \frac{64443098670429961}{6032611833300} \) | = | $2^{-2} \cdot 3^{-8} \cdot 5^{-2} \cdot 13^{-1} \cdot 29^{-4} \cdot 587^{3} \cdot 683^{3}$ |
| Endomorphism ring: | $\Z$ | ||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
| Sato-Tate group: | $\mathrm{SU}(2)$ | ||
| Faltings height: | $1.1907403618847302263271787146\dots$ | ||
| Stable Faltings height: | $1.1907403618847302263271787146\dots$ | ||
BSD invariants
sage: E.rank()
magma: Rank(E);
| |||
| Analytic rank: | $2$ | ||
sage: E.regulator()
magma: Regulator(E);
| |||
| Regulator: | $0.98884997713919495670193802745\dots$ | ||
sage: E.period_lattice().omega()
gp: E.omega[1]
magma: RealPeriod(E);
| |||
| Real period: | $0.50239281289954044390061625978\dots$ | ||
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: | $ 32 $ = $ 2\cdot2\cdot2\cdot1\cdot2^{2} $ | ||
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: | $1$ (rounded) | ||
sage: r = E.rank();
gp: ar = ellanalyticrank(E);
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
| |||
| Special value: | $ L^{(2)}(E,1)/2! $ ≈ $ 3.9743289724048513368048845069 $ | ||
Modular invariants
Modular form 11310.2.a.c
For more coefficients, see the Downloads section to the right.
|
sage: E.modular_degree()
magma: ModularDegree(E);
|
|||
| Modular degree: | 49152 | ||
| $ \Gamma_0(N) $-optimal: | no | ||
| Manin constant: | 1 | ||
Local data
This elliptic curve is semistable. There are 5 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
| $3$ | $2$ | $I_{8}$ | Non-split multiplicative | 1 | 1 | 8 | 8 |
| $5$ | $2$ | $I_{2}$ | Split multiplicative | -1 | 1 | 2 | 2 |
| $13$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
| $29$ | $4$ | $I_{4}$ | Split multiplicative | -1 | 1 | 4 | 4 |
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 | 4.12.0.8 |
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | nonsplit | nonsplit | split | ord | ord | split | ord | ord | ord | split | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | 2 | 2 | 3 | 2 | 2 | 3 | 2 | 2 | 2 | 3 | 2 | 2 | 2 | 2 | 2,2 |
| $\mu$-invariant(s) | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0,0 |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 11310c
consists of 4 curves linked by isogenies of
degrees dividing 4.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{13}) \) | \(\Z/2\Z \times \Z/2\Z\) | Not in database |
| $2$ | \(\Q(\sqrt{-1}) \) | \(\Z/4\Z\) | Not in database |
| $2$ | \(\Q(\sqrt{-13}) \) | \(\Z/4\Z\) | Not in database |
| $4$ | 4.2.3515200.1 | \(\Z/2\Z \times \Z/4\Z\) | Not in database |
| $4$ | \(\Q(i, \sqrt{13})\) | \(\Z/2\Z \times \Z/4\Z\) | Not in database |
| $8$ | 8.0.197706096640000.50 | \(\Z/4\Z \times \Z/4\Z\) | Not in database |
| $8$ | 8.0.489596882944.4 | \(\Z/8\Z\) | Not in database |
| $8$ | Deg 8 | \(\Z/8\Z\) | Not in database |
| $8$ | Deg 8 | \(\Z/6\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/2\Z \times \Z/8\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/2\Z \times \Z/8\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/2\Z \times \Z/8\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/2\Z \times \Z/6\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/12\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/12\Z\) | 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.