Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3-x^2+45095x-895303\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3-x^2z+45095xz^2-895303z^3\) | (dehomogenize, simplify) |
\(y^2=x^3+721525x-56577850\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Infinite order Mordell-Weil generators and heights
$P$ | = | \(\left(359, 7660\right)\) | \(\left(23, 380\right)\) |
$\hat{h}(P)$ | ≈ | $0.32060103722878181786273507925$ | $0.75770762709371853082979161126$ |
Integral points
\( \left(23, 380\right) \), \( \left(23, -404\right) \), \( \left(39, 940\right) \), \( \left(39, -980\right) \), \( \left(65, 1486\right) \), \( \left(65, -1552\right) \), \( \left(179, 3500\right) \), \( \left(179, -3680\right) \), \( \left(219, 4300\right) \), \( \left(219, -4520\right) \), \( \left(359, 7660\right) \), \( \left(359, -8020\right) \), \( \left(839, 24620\right) \), \( \left(839, -25460\right) \), \( \left(1199, 41540\right) \), \( \left(1199, -42740\right) \), \( \left(1983, 87796\right) \), \( \left(1983, -89780\right) \), \( \left(3495, 205228\right) \), \( \left(3495, -208724\right) \), \( \left(7079, 592300\right) \), \( \left(7079, -599380\right) \)
Invariants
Conductor: | \( 56350 \) | = | $2 \cdot 5^{2} \cdot 7^{2} \cdot 23$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-6206747115520000 $ | = | $-1 \cdot 2^{19} \cdot 5^{4} \cdot 7^{7} \cdot 23 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{137927116575}{84410368} \) | = | $2^{-19} \cdot 3^{3} \cdot 5^{2} \cdot 7^{-1} \cdot 19^{3} \cdot 23^{-1} \cdot 31^{3}$ | 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);
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Sato-Tate group: | $\mathrm{SU}(2)$ | |||
Faltings height: | $1.7195478751288706984301810890\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $0.21011349645651392101058493954\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.9759468235041325\dots$ | |||
Szpiro ratio: | $4.000535106263379\dots$ |
BSD invariants
Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $0.24198010734038029894458086554\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.24558424242391178012662917292\dots$ | 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: | $ 228 $ = $ 19\cdot3\cdot2^{2}\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! $ ≈ $ 13.549242306168466146632430137 $ | 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
$\displaystyle 13.549242306 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.245584 \cdot 0.241980 \cdot 228}{1^2} \approx 13.549242306$
Modular invariants
Modular form 56350.2.a.bl
For more coefficients, see the Downloads section to the right.
Modular degree: | 350208 | 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)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $19$ | $I_{19}$ | Split multiplicative | -1 | 1 | 19 | 19 |
$5$ | $3$ | $IV$ | Additive | -1 | 2 | 4 | 0 |
$7$ | $4$ | $I_{1}^{*}$ | Additive | -1 | 2 | 7 | 1 |
$23$ | $1$ | $I_{1}$ | Non-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 \( 1288 = 2^{3} \cdot 7 \cdot 23 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 1287 & 0 \end{array}\right),\left(\begin{array}{rr} 281 & 2 \\ 281 & 3 \end{array}\right),\left(\begin{array}{rr} 967 & 2 \\ 967 & 3 \end{array}\right),\left(\begin{array}{rr} 645 & 2 \\ 645 & 3 \end{array}\right),\left(\begin{array}{rr} 185 & 2 \\ 185 & 3 \end{array}\right),\left(\begin{array}{rr} 1287 & 2 \\ 1286 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[1288])$ is a degree-$413653008384$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1288\Z)$.
Isogenies
This curve has no rational isogenies. Its isogeny class 56350bv consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 8050s1, 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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$3$ | 3.1.32200.1 | \(\Z/2\Z\) | Not in database |
$6$ | 6.0.1335449920000.1 | \(\Z/2\Z \oplus \Z/2\Z\) | 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 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Reduction type | split | ss | add | add | ord | ord | ord | ord | nonsplit | ord | ord | ord | ord | ord | ord |
$\lambda$-invariant(s) | 4 | 2,4 | - | - | 2 | 2 | 2 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 2 |
$\mu$-invariant(s) | 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.