Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2=x^3-x^2+48x+576\) | (homogenize, simplify) |
\(y^2z=x^3-x^2z+48xz^2+576z^3\) | (dehomogenize, simplify) |
\(y^2=x^3+3861x+431514\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Infinite order Mordell-Weil generators and heights
$P$ | = | \(\left(0, 24\right)\) | \(\left(-6, 6\right)\) |
$\hat{h}(P)$ | ≈ | $0.48172545730638140956073549466$ | $1.3417911312662104270789207272$ |
Integral points
\((-6,\pm 6)\), \((0,\pm 24)\), \((2,\pm 26)\), \((16,\pm 72)\), \((21,\pm 102)\), \((32,\pm 184)\), \((209,\pm 3016)\), \((624,\pm 15576)\)
Invariants
Conductor: | \( 103056 \) | = | $2^{4} \cdot 3 \cdot 19 \cdot 113$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-158294016 $ | = | $-1 \cdot 2^{13} \cdot 3^{2} \cdot 19 \cdot 113 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{2924207}{38646} \) | = | $2^{-1} \cdot 3^{-2} \cdot 11^{3} \cdot 13^{3} \cdot 19^{-1} \cdot 113^{-1}$ | 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);
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Sato-Tate group: | $\mathrm{SU}(2)$ | |||
Faltings height: | $0.25411136479007786760520766720\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $-0.43903581576986744181202445426\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.7695426332447073\dots$ | |||
Szpiro ratio: | $2.2775574118825324\dots$ |
BSD invariants
Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $0.63034603038857110539910046087\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $1.3476499753607270212574224898\dots$ | 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);
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Tamagawa product: | $ 8 $ = $ 2^{2}\cdot2\cdot1\cdot1 $ | 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^{(2)}(E,1)/2! $ ≈ $ 6.7958864985751194910426405839 $ | 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
$\displaystyle 6.795886499 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 1.347650 \cdot 0.630346 \cdot 8}{1^2} \approx 6.795886499$
Modular invariants
Modular form 103056.2.a.c
For more coefficients, see the Downloads section to the right.
Modular degree: | 36480 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | yes | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 4 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $4$ | $I_{5}^{*}$ | Additive | -1 | 4 | 13 | 1 |
$3$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
$19$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
$113$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$.
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 17176 = 2^{3} \cdot 19 \cdot 113 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 6329 & 2 \\ 6329 & 3 \end{array}\right),\left(\begin{array}{rr} 4295 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 17175 & 2 \\ 17174 & 3 \end{array}\right),\left(\begin{array}{rr} 8589 & 2 \\ 8589 & 3 \end{array}\right),\left(\begin{array}{rr} 8817 & 2 \\ 8817 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 17175 & 0 \end{array}\right)$.
The torsion field $K:=\Q(E[17176])$ is a degree-$15279500851937280$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/17176\Z)$.
Isogenies
This curve has no rational isogenies. Its isogeny class 103056.c consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 12882.m1, its twist by $-4$.
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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$3$ | 3.1.17176.1 | \(\Z/2\Z\) | Not in database |
$6$ | 6.0.5067177227776.1 | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$8$ | deg 8 | \(\Z/3\Z\) | Not in database |
$12$ | deg 12 | \(\Z/4\Z\) | 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 | 113 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Reduction type | add | nonsplit | ord | ord | ss | ord | ord | split | ord | ord | ord | ord | ord | ord | ord | split |
$\lambda$-invariant(s) | - | 2 | 6 | 2 | 2,2 | 2 | 2 | 3 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 |
$\mu$-invariant(s) | - | 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.