Minimal Weierstrass equation
\(y^2+xy+y=x^3+x^2-411787x-97932871\)
Mordell-Weil group structure
$\Z/{2}\Z$
Torsion generators
\( \left(-429, 214\right) \)
Integral points
\( \left(-429, 214\right) \)
Invariants
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor: | \( 1122 \) | = | $2 \cdot 3 \cdot 11 \cdot 17$ |
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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Discriminant: | $340131399900069888 $ | = | $2^{26} \cdot 3^{13} \cdot 11 \cdot 17^{2} $ |
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
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j-invariant: | \( \frac{7722211175253055152433}{340131399900069888} \) | = | $2^{-26} \cdot 3^{-13} \cdot 11^{-1} \cdot 17^{-2} \cdot 19765777^{3}$ |
Endomorphism ring: | $\Z$ | ||
Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
Sato-Tate group: | $\mathrm{SU}(2)$ | ||
Faltings height: | $2.1270874252021988334929935067\dots$ | ||
Stable Faltings height: | $2.1270874252021988334929935067\dots$ |
BSD invariants
sage: E.rank()
magma: Rank(E);
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Analytic rank: | $0$ | ||
sage: E.regulator()
magma: Regulator(E);
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Regulator: | $1$ | ||
sage: E.period_lattice().omega()
gp: E.omega[1]
magma: RealPeriod(E);
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Real period: | $0.18897855991355678243526076561\dots$ | ||
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
magma: TamagawaNumbers(E);
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Tamagawa product: | $ 52 $ = $ ( 2 \cdot 13 )\cdot1\cdot1\cdot2 $ | ||
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
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Torsion order: | $2$ | ||
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Analytic order of Ш: | $1$ (exact) | ||
sage: r = E.rank();
gp: ar = ellanalyticrank(E);
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
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Special value: | $ L(E,1) $ ≈ $ 2.4567212788762381716583899530 $ |
Modular invariants
For more coefficients, see the Downloads section to the right.
sage: E.modular_degree()
magma: ModularDegree(E);
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Modular degree: | 16224 | ||
$ \Gamma_0(N) $-optimal: | yes | ||
Manin constant: | 1 |
Local data
This elliptic curve is semistable. There are 4 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $26$ | $I_{26}$ | Split multiplicative | -1 | 1 | 26 | 26 |
$3$ | $1$ | $I_{13}$ | Non-split multiplicative | 1 | 1 | 13 | 13 |
$11$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
$17$ | $2$ | $I_{2}$ | Split multiplicative | -1 | 1 | 2 | 2 |
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 |
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.
Iwasawa invariants
$p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 |
---|---|---|---|---|---|---|---|
Reduction type | split | nonsplit | ord | ord | split | ord | split |
$\lambda$-invariant(s) | 4 | 0 | 0 | 0 | 1 | 2 | 1 |
$\mu$-invariant(s) | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 17$ of good reduction are zero.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 1122g
consists of 2 curves linked by isogenies of
degree 2.
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 |
---|---|---|---|
$2$ | \(\Q(\sqrt{33}) \) | \(\Z/2\Z \times \Z/2\Z\) | Not in database |
$4$ | 4.0.610368.2 | \(\Z/4\Z\) | Not in database |
$8$ | Deg 8 | \(\Z/2\Z \times \Z/4\Z\) | Not in database |
$8$ | 8.0.405705964916736.6 | \(\Z/2\Z \times \Z/4\Z\) | Not in database |
$8$ | 8.2.3465933379972272.7 | \(\Z/6\Z\) | Not in database |
$16$ | Deg 16 | \(\Z/8\Z\) | Not in database |
$16$ | Deg 16 | \(\Z/2\Z \times \Z/6\Z\) | 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.