Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-15108x+719937\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-15108xz^2+719937z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-1223775x+521162775\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(117, 735)$ | $0.35945203434585551207031066248$ | $\infty$ |
| $(68, 49)$ | $0.40852802061077884971667741520$ | $\infty$ |
Integral points
\((-128,\pm 735)\), \((-43,\pm 1135)\), \((12,\pm 735)\), \((48,\pm 321)\), \((61,\pm 147)\), \((68,\pm 49)\), \((72,\pm 15)\), \((77,\pm 85)\), \((117,\pm 735)\), \((252,\pm 3585)\), \((656,\pm 16513)\), \((1188,\pm 40719)\)
Invariants
| Conductor: | $N$ | = | \( 29400 \) | = | $2^{3} \cdot 3 \cdot 5^{2} \cdot 7^{2}$ |
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| Discriminant: | $\Delta$ | = | $-74118870000$ | = | $-1 \cdot 2^{4} \cdot 3^{2} \cdot 5^{4} \cdot 7^{7} $ |
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| j-invariant: | $j$ | = | \( -\frac{324179200}{63} \) | = | $-1 \cdot 2^{8} \cdot 3^{-2} \cdot 5^{2} \cdot 7^{-1} \cdot 37^{3}$ |
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) |
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $1.0856958059838634496575911148$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.65478763287514176423441574182$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9257130139200891$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.9346805214555154$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.13752618121267138008474724350$ |
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| Real period: | $\Omega$ | ≈ | $1.0591226221563962349096882538$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 48 $ = $ 2\cdot2\cdot3\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $6.9915403037337709748584987829 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 6.991540304 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.059123 \cdot 0.137526 \cdot 48}{1^2} \\ & \approx 6.991540304\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 64512 |
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| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 |
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Local data at primes of bad reduction
This elliptic curve is not semistable. There are 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $III$ | additive | -1 | 3 | 4 | 0 |
| $3$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $5$ | $3$ | $IV$ | additive | -1 | 2 | 4 | 0 |
| $7$ | $4$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 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 label 14.2.0.a.1, level \( 14 = 2 \cdot 7 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 13 & 2 \\ 12 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 13 & 0 \end{array}\right),\left(\begin{array}{rr} 3 & 2 \\ 3 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[14])$ is a degree-$6048$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/14\Z)$.
The table below list all primes $\ell$ for which the Serre invariants associated to the mod-$\ell$ Galois representation are exceptional.
| $\ell$ | Reduction type | Serre weight | Serre conductor |
|---|---|---|---|
| $2$ | additive | $2$ | \( 1225 = 5^{2} \cdot 7^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 9800 = 2^{3} \cdot 5^{2} \cdot 7^{2} \) |
| $5$ | additive | $14$ | \( 1176 = 2^{3} \cdot 3 \cdot 7^{2} \) |
| $7$ | additive | $32$ | \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 29400do consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 4200be1, 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 |
|---|---|---|---|
| $3$ | 3.1.175.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.214375.1 | \(\Z/2\Z \oplus \Z/2\Z\) | 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 | add | nonsplit | add | add | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 2 | - | - | 2 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| $\mu$-invariant(s) | - | 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.