Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3+112x-4194\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3+112xz^2-4194z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+145773x-196100946\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(17, 43\right) \) | $0.19507735893245370518605013339$ | $\infty$ |
| \( \left(\frac{55}{4}, -\frac{59}{8}\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([17:43:1]\) | $0.19507735893245370518605013339$ | $\infty$ |
| \([110:-59:8]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(615, 11232\right) \) | $0.19507735893245370518605013339$ | $\infty$ |
| \( \left(498, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(17, 43\right) \), \( \left(17, -61\right) \), \( \left(25, 107\right) \), \( \left(25, -133\right) \), \( \left(30, 147\right) \), \( \left(30, -178\right) \), \( \left(56, 394\right) \), \( \left(56, -451\right) \), \( \left(225, 3267\right) \), \( \left(225, -3493\right) \), \( \left(420, 8402\right) \), \( \left(420, -8823\right) \), \( \left(3137, 174139\right) \), \( \left(3137, -177277\right) \)
\([17:43:1]\), \([17:-61:1]\), \([25:107:1]\), \([25:-133:1]\), \([30:147:1]\), \([30:-178:1]\), \([56:394:1]\), \([56:-451:1]\), \([225:3267:1]\), \([225:-3493:1]\), \([420:8402:1]\), \([420:-8823:1]\), \([3137:174139:1]\), \([3137:-177277:1]\)
\((615,\pm 11232)\), \((903,\pm 25920)\), \((1083,\pm 35100)\), \((2019,\pm 91260)\), \((8103,\pm 730080)\), \((15123,\pm 1860300)\), \((112935,\pm 37952928)\)
Invariants
| Conductor: | $N$ | = | \( 130 \) | = | $2 \cdot 5 \cdot 13$ |
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| Minimal Discriminant: | $\Delta$ | = | $-7722894400$ | = | $-1 \cdot 2^{6} \cdot 5^{2} \cdot 13^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{157376536199}{7722894400} \) | = | $2^{-6} \cdot 5^{-2} \cdot 13^{-6} \cdot 5399^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.57702299275763624029845250541$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.57702299275763624029845250541$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0187728994158647$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.206491694063401$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.19507735893245370518605013339$ |
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| Real period: | $\Omega$ | ≈ | $0.63070483322777186758380385874$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 24 $ = $ 2\cdot2\cdot( 2 \cdot 3 ) $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $0.73821739879204444066176003316 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 0.738217399 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.630705 \cdot 0.195077 \cdot 24}{2^2} \\ & \approx 0.738217399\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 144 |
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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 semistable. There are 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 6 |
| $5$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $13$ | $6$ | $I_{6}$ | split multiplicative | -1 | 1 | 6 | 6 |
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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2B | 4.12.0.12 | $12$ |
| $3$ | 3B.1.2 | 3.8.0.2 | $8$ |
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
$\left(\begin{array}{rr} 13 & 24 \\ 852 & 253 \end{array}\right),\left(\begin{array}{rr} 1537 & 24 \\ 1536 & 25 \end{array}\right),\left(\begin{array}{rr} 781 & 24 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 391 & 24 \\ 195 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 24 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 15 & 16 \\ 314 & 335 \end{array}\right),\left(\begin{array}{rr} 1081 & 6 \\ 492 & 73 \end{array}\right),\left(\begin{array}{rr} 521 & 24 \\ 1300 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 24 & 1 \end{array}\right),\left(\begin{array}{rr} 957 & 10 \\ 524 & 101 \end{array}\right)$.
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 |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 1 \) |
| $3$ | good | $2$ | \( 5 \) |
| $5$ | split multiplicative | $6$ | \( 26 = 2 \cdot 13 \) |
| $13$ | split multiplicative | $14$ | \( 10 = 2 \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 130a
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
This elliptic curve is its own minimal quadratic twist.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-1}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | 2.0.4.1-8450.5-a2 |
| $2$ | \(\Q(\sqrt{-3}) \) | \(\Z/6\Z\) | 2.0.3.1-16900.2-a2 |
| $3$ | \(\Q(\sqrt[3]{5})\) | \(\Z/6\Z\) | not in database |
| $4$ | \(\Q(\sqrt{14 +2 \sqrt{65}})\) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{7 -4 i})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\zeta_{12})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.1366875.1 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.29160000.1 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $8$ | 8.0.4569760000.4 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.23134410000.2 | \(\Z/12\Z\) | not in database |
| $8$ | 8.0.87609600.3 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $12$ | 12.0.7652750400000000.2 | \(\Z/6\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/12\Z\) | not in database |
| $18$ | 18.0.284258910904006648725908395200000000.3 | \(\Z/18\Z\) | 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 | nonsplit | ord | split | ord | ord | split | ord | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 2 | 1 | 2 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 1 |
| $\mu$-invariant(s) | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.