Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-747075x-238562750\)
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(homogenize, simplify) |
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\(y^2z=x^3-747075xz^2-238562750z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-747075x-238562750\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-505, 3150)$ | $2.4010118976813182182541467983$ | $\infty$ |
| $(-415, 0)$ | $0$ | $2$ |
Integral points
\((-505,\pm 3150)\), \( \left(-415, 0\right) \), \((2145,\pm 89600)\)
Invariants
| Conductor: | $N$ | = | \( 46800 \) | = | $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $2099266191360000000$ | = | $2^{24} \cdot 3^{6} \cdot 5^{7} \cdot 13^{3} $ |
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| j-invariant: | $j$ | = | \( \frac{988345570681}{44994560} \) | = | $2^{-12} \cdot 5^{-1} \cdot 7^{3} \cdot 13^{-3} \cdot 1423^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.2776216835887139280050708512$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.23044940247766358558983644467$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9543225417743602$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.852810116555073$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.4010118976813182182541467983$ |
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| Real period: | $\Omega$ | ≈ | $0.16284728769757911332100013964$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 2^{2}\cdot2\cdot2^{2}\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.1279862021361601046725404511 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.127986202 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.162847 \cdot 2.401012 \cdot 32}{2^2} \\ & \approx 3.127986202\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 995328 |
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| $ \Gamma_0(N) $-optimal: | no | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{16}^{*}$ | additive | -1 | 4 | 24 | 12 |
| $3$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $5$ | $4$ | $I_{1}^{*}$ | additive | 1 | 2 | 7 | 1 |
| $13$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
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 | 4.6.0.3 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \), index $384$, genus $9$, and generators
$\left(\begin{array}{rr} 1 & 12 \\ 12 & 145 \end{array}\right),\left(\begin{array}{rr} 1537 & 24 \\ 1536 & 25 \end{array}\right),\left(\begin{array}{rr} 1169 & 1536 \\ 1548 & 1271 \end{array}\right),\left(\begin{array}{rr} 738 & 1 \\ 143 & 8 \end{array}\right),\left(\begin{array}{rr} 1039 & 1536 \\ 1040 & 1559 \end{array}\right),\left(\begin{array}{rr} 1 & 24 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 21 & 4 \\ 1460 & 1541 \end{array}\right),\left(\begin{array}{rr} 7 & 24 \\ 492 & 127 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 24 & 1 \end{array}\right),\left(\begin{array}{rr} 781 & 24 \\ 780 & 1 \end{array}\right),\left(\begin{array}{rr} 608 & 1557 \\ 1083 & 86 \end{array}\right)$.
The torsion field $K:=\Q(E[1560])$ is a degree-$2415329280$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1560\Z)$.
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 | $2$ | \( 2925 = 3^{2} \cdot 5^{2} \cdot 13 \) |
| $3$ | additive | $6$ | \( 400 = 2^{4} \cdot 5^{2} \) |
| $5$ | additive | $18$ | \( 1872 = 2^{4} \cdot 3^{2} \cdot 13 \) |
| $13$ | nonsplit multiplicative | $14$ | \( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 46800.d
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 130.a1, its twist by $60$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{65}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-5}) \) | \(\Z/6\Z\) | not in database |
| $4$ | 4.0.9360.2 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-5}, \sqrt{-13})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.2.437400000.2 | \(\Z/6\Z\) | not in database |
| $8$ | 8.4.97742882250000.4 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.370150560000.12 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.2190240000.13 | \(\Z/12\Z\) | 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 | \(\Z/4\Z \oplus \Z/4\Z\) | 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 |
| $18$ | 18.0.1212838019857095034563875819520000000000000.2 | \(\Z/18\Z\) | 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 | add | add | add | ord | ord | nonsplit | ord | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | - | - | 5 | 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 |
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.