Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-2473065x+1496723617\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-2473065xz^2+1496723617z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-3205092267x+69840752351526\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(924, 383)$ | $0.68673488528653162707970776485$ | $\infty$ |
| $(2224, 82543)$ | $3.5708686672430489096992109619$ | $\infty$ |
| $(3631/4, -3631/8)$ | $0$ | $2$ |
Integral points
\( \left(-1416, 47183\right) \), \( \left(-1416, -45767\right) \), \( \left(564, 16493\right) \), \( \left(564, -17057\right) \), \( \left(872, 1423\right) \), \( \left(872, -2295\right) \), \( \left(908, -449\right) \), \( \left(908, -459\right) \), \( \left(924, 383\right) \), \( \left(924, -1307\right) \), \( \left(2224, 82543\right) \), \( \left(2224, -84767\right) \)
Invariants
| Conductor: | $N$ | = | \( 130130 \) | = | $2 \cdot 5 \cdot 7 \cdot 11 \cdot 13^{2}$ |
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| Discriminant: | $\Delta$ | = | $40065410785400$ | = | $2^{3} \cdot 5^{2} \cdot 7^{3} \cdot 11^{2} \cdot 13^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{346553430870203929}{8300600} \) | = | $2^{-3} \cdot 5^{-2} \cdot 7^{-3} \cdot 11^{-2} \cdot 29^{3} \cdot 53^{3} \cdot 457^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.1306870913096867854837139825$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.84821241257891841745697026172$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9755108563325865$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.736339984505729$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.4095261969475827879388139358$ |
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| Real period: | $\Omega$ | ≈ | $0.46857382723798065954229373298$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 48 $ = $ 3\cdot2\cdot1\cdot2\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $13.548490943206862626534079809 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 13.548490943 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.468574 \cdot 2.409526 \cdot 48}{2^2} \\ & \approx 13.548490943\end{aligned}$$
Modular invariants
Modular form 130130.2.a.bq
For more coefficients, see the Downloads section to the right.
| Modular degree: | 2654208 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $5$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $7$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $11$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $13$ | $4$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
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 | 2.3.0.1 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 120120 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 25026 & 41977 \\ 85085 & 65066 \end{array}\right),\left(\begin{array}{rr} 96097 & 36972 \\ 114582 & 101713 \end{array}\right),\left(\begin{array}{rr} 76441 & 36972 \\ 116766 & 101713 \end{array}\right),\left(\begin{array}{rr} 120109 & 12 \\ 120108 & 13 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 27730 & 9243 \\ 96993 & 92392 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 83159 & 0 \\ 0 & 120119 \end{array}\right),\left(\begin{array}{rr} 62050 & 9243 \\ 88413 & 92392 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 120070 & 120111 \end{array}\right),\left(\begin{array}{rr} 104729 & 46202 \\ 118638 & 36973 \end{array}\right)$.
The torsion field $K:=\Q(E[120120])$ is a degree-$257099242143744000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/120120\Z)$.
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 |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 1183 = 7 \cdot 13^{2} \) |
| $3$ | good | $2$ | \( 9295 = 5 \cdot 11 \cdot 13^{2} \) |
| $5$ | split multiplicative | $6$ | \( 26026 = 2 \cdot 7 \cdot 11 \cdot 13^{2} \) |
| $7$ | nonsplit multiplicative | $8$ | \( 18590 = 2 \cdot 5 \cdot 11 \cdot 13^{2} \) |
| $11$ | split multiplicative | $12$ | \( 11830 = 2 \cdot 5 \cdot 7 \cdot 13^{2} \) |
| $13$ | additive | $86$ | \( 770 = 2 \cdot 5 \cdot 7 \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 130130.bq
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 770.a2, its twist by $13$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{14}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{13}) \) | \(\Z/6\Z\) | not in database |
| $4$ | 4.0.114514400.10 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{13}, \sqrt{14})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.542805924375.3 | \(\Z/6\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 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 |
| $18$ | 18.6.4761576343176867701258515460444132483203125.1 | \(\Z/18\Z\) | 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 | ord | split | nonsplit | split | add | ord | ord | ord | ss | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 4 | 4 | 3 | 2 | 3 | - | 2 | 2 | 2 | 2,2 | 2 | 2 | 2 | 2 | 2 |
| $\mu$-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
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.