Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3+24670x+4571452\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3+24670xz^2+4571452z^3\) | (dehomogenize, simplify) |
\(y^2=x^3+31972941x+213189757326\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(8464, 774604\right)\) |
$\hat{h}(P)$ | ≈ | $5.2092423072953919642524435409$ |
Integral points
\( \left(8464, 774604\right) \), \( \left(8464, -783069\right) \)
Invariants
Conductor: | \( 16562 \) | = | $2 \cdot 7^{2} \cdot 13^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-9980869973872616 $ | = | $-1 \cdot 2^{3} \cdot 7^{6} \cdot 13^{9} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{1331}{8} \) | = | $2^{-3} \cdot 11^{3}$ | comment: j-invariant
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
oscar: j_invariant(E)
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Endomorphism ring: | $\Z$ | |||
Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | sage: E.has_cm()
magma: HasComplexMultiplication(E);
| |
Sato-Tate group: | $\mathrm{SU}(2)$ | |||
Faltings height: | $1.7518516556687437753800357788\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $-1.1448154369550654292127561741\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.9357683713670925\dots$ | |||
Szpiro ratio: | $4.549000263849239\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $5.2092423072953919642524435409\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.29501633069865213497272803181\dots$ | comment: Real Period
sage: E.period_lattice().omega()
gp: if(E.disc>0,2,1)*E.omega[1]
magma: (Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E);
|
Tamagawa product: | $ 2 $ = $ 1\cdot1\cdot2 $ | comment: Tamagawa numbers
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
magma: TamagawaNumbers(E);
oscar: tamagawa_numbers(E)
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Torsion order: | $1$ | comment: Torsion order
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
oscar: prod(torsion_structure(E)[1])
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Analytic order of Ш: | $1$ ( rounded) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Special value: | $ L'(E,1) $ ≈ $ 3.0736231024369340455748649359 $ | comment: Special L-value
r = E.rank();
gp: [r,L1r] = ellanalyticrank(E); L1r/r!
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
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BSD formula
$\displaystyle 3.073623102 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.295016 \cdot 5.209242 \cdot 2}{1^2} \approx 3.073623102$
Modular invariants
Modular form 16562.2.a.o
For more coefficients, see the Downloads section to the right.
Modular degree: | 102960 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | yes | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
|
Local data
This elliptic curve is not semistable. There are 3 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $1$ | $I_{3}$ | Non-split multiplicative | 1 | 1 | 3 | 3 |
$7$ | $1$ | $I_0^{*}$ | Additive | -1 | 2 | 6 | 0 |
$13$ | $2$ | $III^{*}$ | Additive | -1 | 2 | 9 | 0 |
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 |
---|---|---|
$3$ | 3Ns | 3.6.0.1 |
$5$ | 5B | 5.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 10920 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 13 \), index $576$, genus $17$, and generators
$\left(\begin{array}{rr} 1261 & 3150 \\ 4410 & 2857 \end{array}\right),\left(\begin{array}{rr} 4201 & 0 \\ 0 & 3361 \end{array}\right),\left(\begin{array}{rr} 3641 & 0 \\ 0 & 3641 \end{array}\right),\left(\begin{array}{rr} 3119 & 0 \\ 0 & 10919 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4620 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4200 & 1 \end{array}\right),\left(\begin{array}{rr} 5461 & 0 \\ 4620 & 5461 \end{array}\right),\left(\begin{array}{rr} 3641 & 4830 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1576 & 6615 \\ 3885 & 5986 \end{array}\right),\left(\begin{array}{rr} 736 & 5775 \\ 7665 & 4831 \end{array}\right),\left(\begin{array}{rr} 6721 & 6720 \\ 4200 & 6721 \end{array}\right),\left(\begin{array}{rr} 10711 & 630 \\ 0 & 10711 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 9360 & 1 \end{array}\right),\left(\begin{array}{rr} 8133 & 3283 \\ 6230 & 3807 \end{array}\right),\left(\begin{array}{rr} 8191 & 4830 \\ 4305 & 6931 \end{array}\right),\left(\begin{array}{rr} 1 & 4368 \\ 3360 & 1 \end{array}\right),\left(\begin{array}{rr} 1561 & 9360 \\ 1560 & 1561 \end{array}\right),\left(\begin{array}{rr} 5041 & 6090 \\ 2310 & 7981 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 3150 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[10920])$ is a degree-$3246202552320$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/10920\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
5.
Its isogeny class 16562.o
consists of 2 curves linked by isogenies of
degree 5.
Twists
The minimal quadratic twist of this elliptic curve is 338.d2, its twist by $-91$.
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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$3$ | 3.1.104.1 | \(\Z/2\Z\) | Not in database |
$4$ | 4.0.968877.1 | \(\Z/3\Z\) | Not in database |
$4$ | 4.2.322959.1 | \(\Z/3\Z\) | Not in database |
$4$ | 4.4.107653.1 | \(\Z/5\Z\) | Not in database |
$6$ | 6.0.1124864.1 | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$8$ | 8.0.938722641129.9 | \(\Z/3\Z \oplus \Z/3\Z\) | Not in database |
$8$ | 8.4.104302515681.1 | \(\Z/15\Z\) | Not in database |
$12$ | deg 12 | \(\Z/4\Z\) | Not in database |
$12$ | deg 12 | \(\Z/6\Z\) | Not in database |
$12$ | deg 12 | \(\Z/6\Z\) | Not in database |
$12$ | deg 12 | \(\Z/10\Z\) | Not in database |
$16$ | deg 16 | \(\Z/3\Z \oplus \Z/15\Z\) | Not in database |
$20$ | 20.0.28917495748885294184346986000000000000000.2 | \(\Z/5\Z\) | 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.
Iwasawa invariants
$p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Reduction type | nonsplit | ord | ord | add | ss | add | ord | ord | ord | ss | ss | ord | ss | ord | ord |
$\lambda$-invariant(s) | 2 | 3 | 1 | - | 1,1 | - | 1 | 1 | 1 | 1,1 | 1,1 | 1 | 1,1 | 1 | 1 |
$\mu$-invariant(s) | 0 | 0 | 0 | - | 0,0 | - | 0 | 0 | 0 | 0,0 | 0,0 | 0 | 0,0 | 0 | 0 |
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.