Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3+x^2-10066953x+12165719607\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3+x^2z-10066953xz^2+12165719607z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-13046771115x+567799515559206\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(1287, 36008\right)\) |
$\hat{h}(P)$ | ≈ | $0.62867335727335982891474884882$ |
Torsion generators
\( \left(1679, -840\right) \)
Integral points
\( \left(1287, 36008\right) \), \( \left(1287, -37296\right) \), \( \left(1679, -840\right) \), \( \left(2255, 29496\right) \), \( \left(2255, -31752\right) \), \( \left(5775, 380088\right) \), \( \left(5775, -385864\right) \)
Invariants
Conductor: | \( 54978 \) | = | $2 \cdot 3 \cdot 7^{2} \cdot 11 \cdot 17$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $1311981684929612218368 $ | = | $2^{30} \cdot 3^{3} \cdot 7^{6} \cdot 11^{3} \cdot 17^{2} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{959024269496848362625}{11151660319506432} \) | = | $2^{-30} \cdot 3^{-3} \cdot 5^{3} \cdot 11^{-3} \cdot 17^{-2} \cdot 97^{3} \cdot 20333^{3}$ | 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: | $2.8645938148526761036648828633\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
||
Stable Faltings height: | $1.8916387403250194511122064916\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $0.62867335727335982891474884882\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
|
Real period: | $0.15325869262121039570021760533\dots$ | 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: | $ 360 $ = $ ( 2 \cdot 3 \cdot 5 )\cdot1\cdot2\cdot3\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: | $2$ | 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) $ ≈ $ 8.6714691139352014976507690827 $ | comment: Special L-value
r = E.rank();
gp: [r,L1r] = ellanalyticrank(E); L1r/r!
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
|
BSD formula
$\displaystyle 8.671469114 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.153259 \cdot 0.628673 \cdot 360}{2^2} \approx 8.671469114$
Modular invariants
Modular form 54978.2.a.bo
For more coefficients, see the Downloads section to the right.
Modular degree: | 4147200 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | yes | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $30$ | $I_{30}$ | Split multiplicative | -1 | 1 | 30 | 30 |
$3$ | $1$ | $I_{3}$ | Non-split multiplicative | 1 | 1 | 3 | 3 |
$7$ | $2$ | $I_0^{*}$ | Additive | -1 | 2 | 6 | 0 |
$11$ | $3$ | $I_{3}$ | Split multiplicative | -1 | 1 | 3 | 3 |
$17$ | $2$ | $I_{2}$ | 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 | 8.6.0.4 |
$3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1848 = 2^{3} \cdot 3 \cdot 7 \cdot 11 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 1450 & 1323 \\ 1365 & 1576 \end{array}\right),\left(\begin{array}{rr} 50 & 539 \\ 693 & 1828 \end{array}\right),\left(\begin{array}{rr} 925 & 1596 \\ 798 & 337 \end{array}\right),\left(\begin{array}{rr} 15 & 322 \\ 1022 & 533 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 1798 & 1839 \end{array}\right),\left(\begin{array}{rr} 1837 & 12 \\ 1836 & 13 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 263 & 0 \\ 0 & 1847 \end{array}\right)$.
The torsion field $K:=\Q(E[1848])$ is a degree-$20437401600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1848\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 54978.bo
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 1122.j3, its twist by $-7$.
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 |
---|---|---|---|
$2$ | \(\Q(\sqrt{33}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$2$ | \(\Q(\sqrt{-7}) \) | \(\Z/6\Z\) | Not in database |
$4$ | 4.0.103488.2 | \(\Z/4\Z\) | Not in database |
$4$ | \(\Q(\sqrt{-7}, \sqrt{33})\) | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$6$ | 6.2.773487981.3 | \(\Z/6\Z\) | Not in database |
$8$ | 8.0.11662935330816.22 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | 8.0.10709766144.8 | \(\Z/12\Z\) | Not in database |
$12$ | deg 12 | \(\Z/3\Z \oplus \Z/6\Z\) | Not in database |
$12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$16$ | deg 16 | \(\Z/8\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
$18$ | 18.0.745046780897063956598099954038014503954531217408.1 | \(\Z/18\Z\) | 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.
Iwasawa invariants
$p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Reduction type | split | nonsplit | ss | add | split | ord | split | ord | ord | ord | ord | ord | ord | ord | ord |
$\lambda$-invariant(s) | 8 | 5 | 1,1 | - | 2 | 1 | 2 | 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 |
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.